Recent zbMATH articles in MSC 14https://www.zbmath.org/atom/cc/142022-01-14T13:23:02.489162ZUnknown authorWerkzeugModel completeness for the Real Field with the Weierstrass \(\wp\) functionhttps://www.zbmath.org/1475.030842022-01-14T13:23:02.489162Z"Bianconi, Ricardo"https://www.zbmath.org/authors/?q=ai:bianconi.ricardoSummary: We prove model completeness for the expansion of the real field by the Weierstrass \(\wp\) function as a function of the variable \(z\) and the parameter (or period) \({\tau}\). We need to existentially define the partial derivatives of the \(\wp\) function with respect to the variable \(z\) and the parameter \({\tau}\). To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass \({\zeta}\) function and the quasi-modular form \(E_{2}\) (we conjecture that these functions are not existentially definable from the functions \(\wp\) alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.Twisted Galois stratificationhttps://www.zbmath.org/1475.030882022-01-14T13:23:02.489162Z"Tomašić, Ivan"https://www.zbmath.org/authors/?q=ai:tomasic.ivanThis paper develops the theory of twisted Galois stratification in order to describe first-order definable sets in the language of difference rings over algebraic closures of finite fields equipped with powers of the Frobenius automorphism. After introducing basic concepts of the theory of difference schemes and their morphisms, as well as the notions of a (normal) Galois stratification \(\mathcal{A}\) on a difference scheme (\(X, \sigma\)) and the Galois formula associated with \(\mathcal{A}\), the author develops difference algebraic geometry (in particular, the theory of generalized difference schemes). The main result of the paper is a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence of this theorem, the author obtains an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes. In addition, the paper presents a number of new results on the category of difference schemes, Babbitt's decomposition, and effective difference algebraic geometry.
Reviewer: Alexander B. Levin (Washington)An algorithm for regular solutions of systems of exp-subanalytic equations and o-minimality of \(\mathbb{R}_{\text{an} , \exp}\)https://www.zbmath.org/1475.030912022-01-14T13:23:02.489162Z"Pawłucki, Wiesław"https://www.zbmath.org/authors/?q=ai:pawlucki.wieslaw"Rożen, Zofia"https://www.zbmath.org/authors/?q=ai:rozen.zofiaIn this paper, the authors consider systems of equations of the form \[(S) : \left\{ \begin{array}{ccc} P_1(x, \exp \lambda_{11}(x),\dots,\exp \lambda_{1m}(x) ) & =& 0\\
\vdots &\vdots &\vdots \\
P_n(x, \exp \lambda_{n1}(x),\dots,\exp \lambda_{nm}(x) ) & =& 0 \end{array} \right.\] where the \(P_i\)'s and \(\lambda_{ij}\)'s are global subanalytic \(C^1\) functions, and the unknown \(x\) varies in \(\mathbb R^n\). The article aims to present an algorithm to reduce a system (S) into finitely many systems of subanalytic equations, by means of restrictions and smooth exp-subanalytic change of variables.
The interest of such reduction is justified by a general result of [\textit{J.-M. Lion}, J. Symb. Log. 67, No. 4, 1616--1622 (2002; Zbl 1042.03030)], which deduces o-minimality from a simple finiteness condition (C). Applied in the case of the expansion of the structure \(\mathbb R_{an}\) of global subanalytic sets by the exponential function, the condition (C) becomes to show the finiteness of the number of regular solutions of a system (S) (with affine \(\lambda_{ij}\)'s). For that matter, the authors claim their algorithm allows, together with Lion's result, to recover the o-minimality of the structure \(\mathbb R_{an,\exp}\), which was a breakthrough result in real analytic geometry (see [\textit{L. van den Dries} and \textit{C. Miller}, Isr. J. Math. 85, No. 1--3, 19--56 (1994; Zbl 0823.03017); \textit{A. J. Wilkie}, J. Am. Math. Soc. 9, No. 4, 1051--1094 (1996; Zbl 0892.03013); \textit{L. van den Dries} et al., Ann. Math. (2) 140, No. 1, 183--205 (1994; Zbl 0837.12006); \textit{J.-M. Lion} and \textit{J.-P. Rolin}, Ann. Inst. Fourier 47, No. 3, 859--884 (1997; Zbl 0873.32004)]).
The reduction is mostly based on subanalytic cell decomposition and an intensive use of the preparation theorem for subanalytic functions (see [\textit{J.-M. Lion} and \textit{J.-P. Rolin}, Ann. Inst. Fourier 47, No. 3, 859--884 (1997; Zbl 0873.32004); \textit{A. Parusiński}, Ann. Sci. Éc. Norm. Supér. (4) 27, No. 6, 661--696 (1994; Zbl 0819.32007)]).
Reviewer: Olivier Le Gal (Le Bourget-du-Lac)Projective embeddings of \(\overline{M}_{0,n}\) and parking functionshttps://www.zbmath.org/1475.050052022-01-14T13:23:02.489162Z"Cavalieri, Renzo"https://www.zbmath.org/authors/?q=ai:cavalieri.renzo"Gillespie, Maria"https://www.zbmath.org/authors/?q=ai:gillespie.maria-monks"Monin, Leonid"https://www.zbmath.org/authors/?q=ai:monin.leonidSummary: The moduli space \(\overline{M}_{0,n}\) may be embedded into the product of projective spaces \(\mathbb{P}^1\times\mathbb{P}^2\times\cdots \times\mathbb{P}^{n-3}\), using a combination of the Kapranov map \(|\psi_n|:\overline{M}_{0,n}\to\mathbb{P}^{n-3}\) and the forgetful maps \(\pi_i:\overline{M}_{0,i}\to\overline{M}_{0,i-1}\). We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height \(n-3\). We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in \(\mathbb{A}^2\times\mathbb{A}^3\cdots\times\mathbb{A}^{n-2})\) is equal to \((2(n-3)-1)!!=(2n-7)(2n-9)\cdots(5)(3)(1)\). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.
Editorial remark: This is an extended version of [\textit{R. Cavalieri} et al., Sémin. Lothar. Comb. 84B, 84B.32, 12 p. (2020; Zbl 1447.05003)].Cyclic Demazure modules and positroid varietieshttps://www.zbmath.org/1475.051702022-01-14T13:23:02.489162Z"Lam, Thomas"https://www.zbmath.org/authors/?q=ai:lam.thomas-fThe classical Borel-Weil theorem identifies the global sections \(\Gamma (G/B, L_{\lambda})\) of a line bundle on a flag variety with the irreducible highest weight representation \(V(\lambda)\). When the same line bundle is restricted to a Schubert variety \(X_{\omega}\) the global sections \(\Gamma (X_{\omega}, L_{\lambda})\) can be identified with the Demazure module \(V_{\omega}(\lambda)\). In this paper, the author studies the global sections \(\Gamma (\Pi_f , \mathcal{O}(d))\) of a line bundle on a positroid subvariety \(\Pi_f\) of the Grassmannian \(Gr(k, n)\). Positroid varieties are certain intersections of cyclically rotated Schubert varieties in the Grassmannian. They were introduced in \textit{A. Postnikov}'s work on the totally nonnegative Grassmannian [``Total positivity, Grassmannians, and networks'', Preprint, \url{arXiv:math/0609764}], and subsequently studied in algebro-geometric terms by \textit{A. Knutson} et al. [Compos. Math. 149, No. 10, 1710--1752 (2013; Zbl 1330.14086)].
Each graded piece of the homogeneous coordinate ring of a positroid variety is the intersection of cyclically rotated (rectangular) Demazure modules, which the author calls the cyclic Demazure module. In this paper, he shows that the cyclic Demazure module has a canonical basis, and define the cyclic Demazure crystal.
Reviewer: Cenap Özel (İzmir)Bumpless pipe dreams and alternating sign matriceshttps://www.zbmath.org/1475.051722022-01-14T13:23:02.489162Z"Weigandt, Anna"https://www.zbmath.org/authors/?q=ai:weigandt.anna-eSummary: In their work on the infinite flag variety, \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula.
\textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono [loc. cit.].Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoofhttps://www.zbmath.org/1475.110022022-01-14T13:23:02.489162Z"Serre, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:serre.jean-pierreThe present book is a masterpiece. As it is well known, Fernando Gouveas's handwritten notes from the course that the great Jean-Pierre Serre (Abel Prize, 2003) offered at Harvard in 1985 on the topic ``Number of points of curves over finite fields'' have been an invaluable text and an essential source of inspiration for mathematicians ever since.
Even though a few attempts had been made to type up these notes, none had worked until 2017 when a group of mathematicians commenced working systematically in order to produce a literal copy of Gouvea's notes, from when he attended Serre's class at Harvard in 1985.
In 2018 the resulting text in \TeX\ was given to Jean-Pierre Serre to check the proofs and typography, revise it if needed and add a few complements in various places. The outcome is the present book. Serre acknowledges the numerous people who contributed to this book, mentioning especially F. Q. Gouvea, A. Bassa, E. Howe, E. Lorenzo Garcia, C. Ritzenthaler, R. Schoof and J. Oesterlé.
After a very useful Introduction, the book is separated into two parts. The first part entitled ``\(g\) small'' features the three sections: ``Refinements of Weil's bound'', ``The case of genus 2'', and ``The case of \(g=3\)''. Subsequently, the second part entitled ``\(g\) large'' features the three sections: ``General results'', ``Optimization in the explicit formulas'', and ``The case \(q=2\)''.
The book constitutes an extremely valuable piece of mathematics, that undoubtedly belongs to the bookshelf of every mathematician and must surely be part of all library collections of Universities and Research Institutes.
Reviewer: Michael Th. Rassias (Zürich)Overview of the work of Kumar Murtyhttps://www.zbmath.org/1475.110032022-01-14T13:23:02.489162Z"Akbary, Amir"https://www.zbmath.org/authors/?q=ai:akbary.amir"Gun, Sanoli"https://www.zbmath.org/authors/?q=ai:gun.sanoli"Murty, M. Ram"https://www.zbmath.org/authors/?q=ai:murty.maruti-ramSummary: The role of the scholar in society is foundational for the growth of human civilization. In fact, one could argue that without the scholar, civilizations crumble. The transmission of knowledge from generation to generation, to take what is essential from the past, to transform it into a new shape and arrangement relevant to the present and to stimulate future students to add to this knowledge is the primary role of the teacher. Spanning more than four decades, Kumar Murty has been the model teacher and researcher, working in diverse areas of number theory and arithmetic geometry, expanding his contributions to meet the challenges of the digital age and training an army of students and postdoctoral fellows who will teach the future generations. On top of this, he has also given serious attention to how mathematics and mathematical thought can be applied to dealing with large-scale economic problems and the emergence of ``smart villages''. We will not discuss this latter work here, nor his other work in the field of Indian philosophy. We will only focus on giving a synoptic overview of his major contributions to mathematics.
For the entire collection see [Zbl 1403.11002].Arithmetic, geometry, cryptography and coding theory, AGC2T, 17th international conference, Centre International de Rencontres Mathématiques, Marseilles, France, June 10--14, 2019https://www.zbmath.org/1475.110042022-01-14T13:23:02.489162ZPublisher's description: This volume contains the proceedings of the 17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory (AGC2T-17), held from June 10--14, 2019, at the Centre International de Rencontres Mathématiques in Marseille, France. The conference was dedicated to the memory of Gilles Lachaud, one of the founding fathers of the AGC2T series.
Since the first meeting in 1987 the biennial AGC2T meetings have brought together the leading experts on arithmetic and algebraic geometry, and the connections to coding theory, cryptography, and algorithmic complexity. This volume highlights important new developments in the field.
Readership
Graduate students and research mathematicians interested in explicit methods in arithmetic and algebraic geometry with applications to coding theory, cryptography and algorithmic complexity.
The articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1410.11003].Tropical sequences associated with Somos sequenceshttps://www.zbmath.org/1475.110252022-01-14T13:23:02.489162Z"Bykovskiĭ, Viktor Alekseevich"https://www.zbmath.org/authors/?q=ai:bykovskii.v-a"Romanov, Mark Anatol'evich"https://www.zbmath.org/authors/?q=ai:romanov.mark-anatolevich"Ustinov, Alekseĭ Vladimirovich"https://www.zbmath.org/authors/?q=ai:ustinov.aleksei-vSummary: Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-\(k\) sequence for \(k=4,5,6,7\) was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for \(k=6\) via the Klein sigma function on hyperelliptic curve of genus 2. It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For \(k=4,5,6,7\), the Somos sequences are Laurent polynomials of \(k\) initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables.
Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free terms are written explicitly.\(F\)-sets and finite automatahttps://www.zbmath.org/1475.110382022-01-14T13:23:02.489162Z"Bell, Jason"https://www.zbmath.org/authors/?q=ai:bell.jason-p"Moosa, Rahim"https://www.zbmath.org/authors/?q=ai:moosa.rahim-nThe notion of a \(k\)-automatic subset of \(\mathbb{N}\) (that is, a subset \(S\subset\mathbb{N}\) with the property that there is a finite automaton which accepts exactly the words arising as base \(k\) expansions of elements of \(S\)) is extended here to an \(F\)-automatic subset of a finitely generated abelian group \(\Gamma\) equipped with an endomorphism \(F\). An \(F\)-subset means a finite union of finite sets of sums of elements of \(\Gamma\), of \(F\)-invariant subgroups in \(\Gamma\), and of \(F\)-cycles in \(\Gamma\) (subsets of the form \(\{\gamma+F^\delta\gamma+F^{2\delta}\gamma+\cdots +F^{\ell\delta}\gamma\}\) with \(\ell,\delta\in\mathbb{N}\) and \(\gamma\in\Gamma\)). \(F\)-automaticity is also defined via finite-state automata and accepted words. The main results (Theorems 4.2, 6.9 and 7.4) prove automaticity of \(F\)-subsets under some mild conditions. The results are a natural generalisations of Derksen's analog of the Skolem-Mahler-Lech theorem [\textit{H. Derksen}, Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)] to the Mordell-Lang context studied by \textit{R. Moosa} and \textit{T. Scanlon} [Am. J. Math. 126, No. 3, 473--522 (2004; Zbl 1072.03020)].
Reviewer: Thomas B. Ward (Leeds)Diagonal genus 5 curves, elliptic curves over \(\mathbb{Q}(t)\), and rational Diophantine quintupleshttps://www.zbmath.org/1475.110482022-01-14T13:23:02.489162Z"Stoll, Michael"https://www.zbmath.org/authors/?q=ai:stoll.michaelIn the paper under review the author studies two problems from arithmetic algebraic geometry. Firstly, he considers the question how to find all rational points on a ``diagonal'' cure \(C\) of genus \(5\) over \(\mathbb Q\), that is a curve obtained as the smooth intersection of three diagonal quadrics in \(\mathbb P^4\). Secondly the author considers the question how one can find generators of the Mordell-Weil group of an elliptic curve over the rational function field \(\mathbb Q(t)\).
The author provides for each problem an algorithm and applies this algorithm to questions related to rational Diophantine quintuples. In particular, a quintuple \((a_1,a_2,a_3,a_4,a_5)\in (\mathbb Q^*)^5\) is called Diophantine, if \(a_ia_j+1\) is a square for all \(1\leq i<j\leq 5\). The two main theorems, regarding rational Diophantine quintuples proved in this paper are:
\begin{itemize}
\item If \(t\neq 0,\pm 1,\pm 1/2,\pm 1/3,\pm 1/4\) and if \((a_1,a_2,a_3,a_4)=(t-1,t+1,4t,4t(t^2-1))\), then the only possible \(a_5\in \mathbb Q\) such that \((a_1,a_2,a_3,a_4,a_5)\) is a Diophantine quintuple is \[a_5=\frac{4t(2t-1)(4t^2-2t-1)(4t^2+2t-1)(8t^2-1)}{(64t^6-80t^4+16t^2-1)^2}\]
\item If \((1,3,8,120,z)\) is a rational Diophantine quintuple, then \(z=\frac{777480}{8288641}\).
\end{itemize}
Reviewer: Volker Ziegler (Salzburg)Patching over Berkovich curves and quadratic formshttps://www.zbmath.org/1475.110562022-01-14T13:23:02.489162Z"Mehmeti, Vlerë"https://www.zbmath.org/authors/?q=ai:mehmeti.vlereThe field patching technique was first introduced in [\textit{D. Harbater} and \textit{J. Hartmann}, Isr. J. Math. 176, 61--107 (2010; Zbl 1213.14052)]. It was subsequently developed further in [\textit{D. Harbater} et al., Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)] with applications to the period-index and \(u\)-invariant problems over function fields of curves over complete discretely valued fields. Since then it has been successful in dealing with a host of other arithmetic problems, among others, local-global principles for torsors under linear algebraic groups over such fields (see [\textit{D. Harbater} et al., Am. J. Math. 137, No. 6, 1559--1612 (2015; Zbl 1348.11036)].
Let \(F\) be the function field of a curve over a complete discretely valued field. In the Harbater-Hartmann-Krashen setting, local-global principles for (certain) homogenous varieties and torsors under linear algebraic groups over \(F\) are obtained with respect to overfields coming from (not necessarily closed) points of the special fiber of a two-dimensional normal model of \(F\) over the ring of integers of the complete field, and are then translated to local-global principles with respect to discrete valuations of \(F\) (whenever possible).
The paper under review extends the field patching technique to function fields of analytic curves. Let \(k\) be a complete ultrametric valued field (not necessarily complete with respect to a discrete valuation). Let \(C/k\) be a normal irreducible projective curve and \(F\) be its function field. One may regard \(F\) as the sheaf \(\mathscr{M}\) of meromorphic functions on the Berkovich analytification \(C^{an}\). For \(x \in C^{an}\), let \(\mathscr{M}_{x}\) be the fraction field of \(\mathcal{O}_{C^{an},x}\). Among other technicalities, by obtaining analogues of simultaneous factorization properties for connected rational algebraic groups considered by Harbater-Hartmann-Krashen in this setting, the author shows the following local-global principle:
Theorem. Let \(G/F\) be a connected, rational linear algebraic group acting transitively on field-valued points of a variety \(X/F\) (i.e., \(G(L)\) acts transitively on \(X(L)\) for all field extensions \(L/F\)). Let \(V(F)\) be the set of non-trivial rank one valuations extended either from that on \(k\) or which are trivial when restricted to \(k\). The the following local-global principles hold:
\begin{itemize}
\item \(X(F) \neq \varnothing \iff X(\mathscr{M}_{x}) \neq \varnothing\) for all \(x \in C^{an}\).
\item If \(F\) is perfect or \(X\) is smooth, then
\[
X(F) \neq \varnothing \iff X(F_{v}) \neq \varnothing \ \textrm{for all completions of} \ F \ \textrm{with respect to} \ v \in V(F) .
\]
\end{itemize}
This extends the local-global principle of [Harbater, Zbl 1259.12003] to this setting. Note that Harbater-Hartmann-Krashen need the base field \(k\) to be complete with respect to a discrete valuation. Another difference lies in the overfields \(\mathscr{M}_{x}\) considered by the author. She shows that they contain the ones considered by Harbater-Hartmann-Krashen. As a consequence, the author also recovers their aforementioned result.
As an application, the author provides an upper bound to the so called ``strong'' \(u\)-invariant. Recall that the \(u\)-invariant of a field \(k\) is the smallest positive integer \(u(k)\) such that all quadratic forms with number of variables greater than \(u(k)\) has a non-trivial solution. One does not know how the \(u\)-invariant behaves with respect to transcendental field extensions (or even finite degree field extensions). Typically, for nice fields \(k\), one expects that if \(u(k) = n\), then \(u(k(x)) = 2n\). The notion of strong \(u\)-invariant captures this expectation.
The strong \(u\)-invariant \(u_{s}(k)\) (if it exists) is the smallest number \(m\) such that \(u(E) \leq m\) for all finite field extensions \(E/k\) and \(\frac{1}{2}u(E) \leq m\) for all finitely generated field extensions \(E/k\) of transcendence degree \(1\).
Let \(k\) be a complete non-Archimedean valued field with residue field \(\widetilde{k}\) such that \(\textrm{char}(\widetilde{k}) \neq 2\). Denote the divisible closure of the value group \(|k^{\times}|\) by \(\sqrt{|k^{\times}|}\). As an application of the above local-global principle, together with bounds for the strong \(u\)-invariant of the over fields \(\mathscr{M}_{x}\), the author obtains the following upper bound for \(u_{s}(k)\) in terms of \(u_{s}(\widetilde{k})\).
Theorem. Let \(k\) be a complete non-Archimedean valued field with residue field \(\widetilde{k}\) such that \(\textrm{char}(\widetilde{k}) \neq 2\).
\begin{itemize}
\item If \(\textrm{dim}_{\mathbb{Q}}\sqrt{|k^{\times}|} = n\), then \(u_{s}(k) \leq 2^{n+1}u_{s}(\widetilde{k})\).
\item If \(|k^{\times}|\) is a free \(\mathbb{Z}\)-module with \(\textrm{rank}_{\mathbb{Z}}|k^{\times}| = n\), then \(u_{s}(k) \leq 2^{n}u_{s}(\widetilde{k})\).
\end{itemize}
Reviewer: Saurabh Gosavi (Piscataway)A higher weight analogue of Ogg's theorem on Weierstrass pointshttps://www.zbmath.org/1475.110672022-01-14T13:23:02.489162Z"Dicks, Robert"https://www.zbmath.org/authors/?q=ai:dicks.robertQuadratic periods of meromorphic forms on punctured Riemann surfaceshttps://www.zbmath.org/1475.110712022-01-14T13:23:02.489162Z"Eskandari, Payman"https://www.zbmath.org/authors/?q=ai:eskandari.paymanSummary: We give three proofs of a relation involving classical and quadratic periods of meromorphic differentials on a punctured elliptic curve. The first proof is based on an old argument of \textit{R. C. Gunning} (ed.) [Problems in analysis. A symposium in honor of Salomon Bochner. Princeton, NJ: Princeton University Press (1970; Zbl 0208.00301)]. The second proof considers how quadratic periods vary in the Legendre family of elliptic curves. The final proof exploits connections to the Hodge theory of the fundamental group and is suitable for generalization to arbitrary Riemann surfaces. The obstacle for such generalization is a lack of a simple description of the Hodge filtration on the space of iterated integrals of length \(\le 2\) on a punctured Riemann surface of arbitrary genus in terms of meromorphic differentials.
For the entire collection see [Zbl 1403.11002].Overconvergent Eichler-Shimura isomorphisms for unitary Shimura curves over totally real fieldshttps://www.zbmath.org/1475.110812022-01-14T13:23:02.489162Z"Barrera, Daniel"https://www.zbmath.org/authors/?q=ai:barrera.daniel"Gao, Shan"https://www.zbmath.org/authors/?q=ai:gao.shanThe \(p\)-adic Eichler-Shimura isomorphism, introduced by Faltings, describes the space of \(p\)-modular forms, defined as global section of certain automorphic bundle, using the cohomology of certain local systems. This construction works for modular forms of integral weight and can be used, for example, to build Galois representation from modular forms.
Modular forms of integral weight can be interpolated by \(p\)-adic families of modular forms, and the theory of modular symbols introduced by Stevens provides a similar interpolation for the cohomology of the aforementioned local systems. One can then hope to interpolate the Eichler-Shimira isomorphisms. This has been done, at least over some subset of the weight space, by Andreatta, Iovita, and Stevens [\textit{F. Andreatta} et al., J. Inst. Math. Jussieu 14, No. 2, 221--274 (2015; Zbl 1379.11062)].
Since both the theory of \(p\)-adic families of modular forms and of modular symbols exist for a lot of Shimura varieties, a natural problem is to construct families of Eichler-Shimura isomorphisms for Shimura varieties other than the modular curve. In their previous work ``Overconvergent Eichler-Shimura isomorphisms for quaternionic modular forms over \(\mathbb{Q}\)'' [Int. J. Number Theory 13, No. 10, 2687--2715 (2017; Zbl 1428.11101)], the authors consider the case of Shimura curves associated to a quaternion algebra over \(\mathbb{Q}\). In the paper at hand they generalize this construction to Shimura curves associated to a quaternion algebra over a totally real field.
Reviewer: Riccardo Brasca (Paris)Noncommutative geometry of groups like \(\Gamma_0(N)\)https://www.zbmath.org/1475.110992022-01-14T13:23:02.489162Z"Plazas, Jorge"https://www.zbmath.org/authors/?q=ai:plazas.jorgeSummary: We show that the Connes-Marcolli \(\mathrm{GL}_2\)-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the \(\mathrm{GL}_2\)-system is implemented by Conway's distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine.Effective bounds for Huber's constant and Faltings's delta functionhttps://www.zbmath.org/1475.111012022-01-14T13:23:02.489162Z"Avdispahić, Muharem"https://www.zbmath.org/authors/?q=ai:avdispahic.muharemSummary: By a closer inspection of the Friedman-Jorgenson-Kramer algorithm related to the prime geodesic theorem on cofinite Fuchsian groups of the first kind, we refine the constants therein. The newly obtained effective upper bound for Huber's constant is in the modular surface case approximately 74000-times smaller than the previously claimed one. The degree of reduction in the case of an upper bound for Faltings's delta function ranges from \(10^8\) to \(10^{16}\).Cohomology of \(p\)-adic Stein spaceshttps://www.zbmath.org/1475.111102022-01-14T13:23:02.489162Z"Colmez, Pierre"https://www.zbmath.org/authors/?q=ai:colmez.pierre"Dospinescu, Gabriel"https://www.zbmath.org/authors/?q=ai:dospinescu.gabriel"Nizioł, Wiesława"https://www.zbmath.org/authors/?q=ai:niziol.wieslawaLet \(\mathcal{O}_K\) be a discrete valuation ring of mixed characteristic \((0,p)\), with residue field \(k\) and fraction field \(K\). Let \(C\) be the complete algebraic closure of \(K\).
The authors explain how to compute the pro-étale cohomology of the analytic space \(X_C\) associated to a ``semistable Stein weak formal scheme'' \(X\) over \(\mathcal{O}_K\). The main theorem is that \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) can be determined by the following objects:
-- (some part of) the overconvergent Hyodo-Kato cohomology of the reduction \(X \otimes_{\mathcal{O}_K} k\),
-- closed \(r\)-forms on \(X_C\), and
-- the de Rham cohomology \(H^r_{\mathrm{DR}}(X_C)\).
Indeed, \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) is the pullback of a diagram of the form
\[
\left(H_{\mathrm{HK}}^{r}(X_k)\otimes \mathbb{B}_{\mathrm{st}}\right)^{\substack{N=0\\
\varphi=p^r}} \to H^r_{\mathrm{DR}}(X_C) \leftarrow \Omega^{r}(X_C)^{d=0}
\]
The pro-étale cohomology and the above mentioned gadgets are related by the syntomic cohomology. Via the period morphism, the pro-étale cohomology and syntomic cohomology can be identified after a truncation. The authors then introduce a Bloch-Kato type syntomic cohomology, which is more concrete, and prove the two syntomic theories are isomorphic.
The theorem is used to calculate the étale and pro-étale cohomology of Drinfeld half-spaces.
Reviewer: Dingxin Zhang (Beijing)Torsion points with multiplicatively dependent coordinates on elliptic curveshttps://www.zbmath.org/1475.111122022-01-14T13:23:02.489162Z"Barroero, Fabrizio"https://www.zbmath.org/authors/?q=ai:barroero.fabrizio"Sha, Min"https://www.zbmath.org/authors/?q=ai:sha.min|sha.min.1The authors prove several results regarding the multiplicative dependence of coordinates of torsion points on elliptic curves defined over a number field. In particular, they prove that for an elliptic curve \(E\) defined over a number field, there are only finitely many torsion points of \(E\) whose coordinates (with respect to some fixed embedding) are multiplicatively independent.
However, the authors prove some more general results as well, including proving the multiplicative independence of fixed (multiplicatively independent) rational functions of the coordinates. In addition, the authors find an effective bound for the order of the torsion points with multiplicatively dependent coordinates, in the case that the curve \(E\) has complex multiplication.
Reviewer: David McKinnon (Waterloo)Torsion of elliptic curves with rational \(j\)-invariant defined over number fields of prime degreehttps://www.zbmath.org/1475.111132022-01-14T13:23:02.489162Z"Gužvić, Tomislav"https://www.zbmath.org/authors/?q=ai:guzvic.tomislavSummary: Let \([K:\mathbb{Q}]=p\) be a prime number and let \(E/K\) be an elliptic curve with \(j(E)\in\mathbb{Q}\). We determine the all possibilities for \(E(K)_{\mathrm{tors}}\). We obtain these results by studying Galois representations of \(E\) and of its quadratic twists.On Ceva points of (almost) equilateral triangleshttps://www.zbmath.org/1475.111142022-01-14T13:23:02.489162Z"Laflamme, Jeanne"https://www.zbmath.org/authors/?q=ai:laflamme.jeanne"Lalín, Matilde"https://www.zbmath.org/authors/?q=ai:lalin.matilde-nThe paper under review proves the infinitude of Ceva points on equilateral and almost equilateral triangles that are also rational. The pleasing central idea of the proof is to construct a parameter space that turns out to be an elliptic surface of positive rank.
A Ceva point of a triangle is a point whose three cevians have rational length. The cevians of a point \(P\) are the three line segments that join a vertex of the triangle to the opposite side, and whose corresponding lines contain \(P\). An almost equilateral triangle is a triangle whose side lengths are three consecutive integers. A triangle is rational if and only if its side lengths are rational.
In fact, the authors prove more than this, namely, that all but finitely many cevians of rational length contain infinitely many Ceva points.
Reviewer: David McKinnon (Waterloo)Reductions of points on algebraic groups. IIhttps://www.zbmath.org/1475.111192022-01-14T13:23:02.489162Z"Bruin, Peter"https://www.zbmath.org/authors/?q=ai:bruin.peter"Perucca, Antonella"https://www.zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus over a number field \(K\), and let \(m\geqslant 2\) be a square-free integer. If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \((\alpha\bmod\mathfrak{p})\) is well defined and has order coprime to \(m\). This set admits a natural density, which we are able to express as a finite sum of products of \(\ell\)-adic integrals, where \(\ell\) varies in the set of prime divisors of \(m\). We deduce that the density is a rational number, whose denominator is bounded (up to powers of \(m)\) in a very strong sense. This extends the results of the paper \textit{Reductions of points on algebraic groups} by \textit{D. Lombardo} and the second author, where the case \(m\) prime is established [Part I, J. Inst. Math. Jussieu 20, No. 5, 1637--1669 (2021; Zbl 1475.11122)].Bounds of the rank of the Mordell-Weil group of Jacobians of hyperelliptic curveshttps://www.zbmath.org/1475.111202022-01-14T13:23:02.489162Z"Daniels, Harris B."https://www.zbmath.org/authors/?q=ai:daniels.harris-b"Lozano-Robledo, Álvaro"https://www.zbmath.org/authors/?q=ai:lozano-robledo.alvaro"Wallace, Erik"https://www.zbmath.org/authors/?q=ai:wallace.erikLet \(C/\mathbb{Q}\) be a hyperelliptic curve given by a model \(y^2 = f(x)\), with \(f(x) \in \mathbb{Q}[x]\), and let \(J/\mathbb{Q}\) be its Jacobian. The Mordell-Weil theorem states that \(J(\mathbb{Q})\) is a finitely generated abelian group and hence \(J(\mathbb{Q})\) decomposes as a direct sum \(J(\mathbb{Q})_{\text{tors}} \oplus \mathbb{Z}^{R_{J(\mathbb{Q})}}\), where \(J(\mathbb{Q})_{\text{tors}}\) is the subgroup of torsion elements and \(R_{J(\mathbb{Q})}\) is the rank of \(J(\mathbb{Q})\). During the last decades a great amount of research has gone into finding bounds of \(R_{J(\mathbb{Q})}\) in terms of invariants of \(C\). In this article the authors give families of examples of hyperelliptic curves \(C \colon y^2 = f(x)\) defined over \(\mathbb{Q}\), with \(f(x)\) of degree \(p\), where \(p\) is a Sophie Germain prime, such that \(R_{J(\mathbb{Q})}\) is bounded by the genus of \(C\) and the two-rank of the class group of the cyclic field defined by \(f(x)\). They further exhibit examples where the given bound is sharp. This extends work of \textit{D. Shanks} [Math. Comput. 28, 1137--1152 (1974; Zbl 0307.12005)] and
\textit{L. C. Washington} [Math. Comput. 48, 371--384 (1987; Zbl 0613.12002)] where a similar bound is given for the rank of certain elliptic curves.
Reviewer: Ana María Botero (Regensburg)On the periods of abelian varietieshttps://www.zbmath.org/1475.111212022-01-14T13:23:02.489162Z"Gross, Benedict H."https://www.zbmath.org/authors/?q=ai:gross.benedict-hSummary: In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker's limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.Reductions of points on algebraic groupshttps://www.zbmath.org/1475.111222022-01-14T13:23:02.489162Z"Lombardo, Davide"https://www.zbmath.org/authors/?q=ai:lombardo.davide-m"Perucca, Antonella"https://www.zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus defined over a number field \(K\). Fix some prime number \(\ell \). If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \(( \alpha \bmod \mathfrak{p})\) is well-defined and has order coprime to \(\ell \). This set admits a natural density. By refining the method of \textit{R. Jones} and \textit{J. Rouse} [Proc. Lond. Math. Soc. (3) 100, No. 3, 763--794 (2010; Zbl 1244.11057)], we can express the density as an \(\ell \)-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of \(\ell )\) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.On the bad reduction of certain \(U(2,1)\) Shimura varietieshttps://www.zbmath.org/1475.111232022-01-14T13:23:02.489162Z"De Shalit, Ehud"https://www.zbmath.org/authors/?q=ai:de-shalit.ehud"Goren, Eyal Z."https://www.zbmath.org/authors/?q=ai:goren.eyal-zSummary: Let \(E\) be a quadratic imaginary field, and let \(p\) be a prime which is inert in \(E\). We study three types of Picard modular surfaces in positive characteristic \(p\) and the morphisms between them. The first Picard surface, denoted S, parametrizes triples \((A,\phi,\iota)\) comprised of an abelian threefold Awith an action \(\iota\) of the ring of integers \(\mathcal{O}_E\), and a principal polarization \(\phi\). The second surface, \(S_0(p)\), parametrizes, in addition, a suitably restricted choice of a subgroup \(H\subset A[p]\) of rank \(p^2\). The third Picard surface, \(\widetilde{S}\), parametrizes triples \((A,\psi,\iota)\) similar to those parametrized by \(S\), but where \(\psi\) is a polarization of degree \(p^2\). We study the components, singularities and naturally defined stratifications of these surfaces, and their behavior under the morphisms. A particular role is played by a foliation we define on the blowup of Sat its superspecial points.
For the entire collection see [Zbl 1403.11002].Incoherent definite spaces and Shimura varietieshttps://www.zbmath.org/1475.111242022-01-14T13:23:02.489162Z"Gross, Benedict H."https://www.zbmath.org/authors/?q=ai:gross.benedict-hSummary: In this paper, we define incoherent definite quadratic spaces over totally real number fields and incoherent definite Hermitian spaces over CM fields. We use the neighbors of these spaces to study the local points of orthogonal and unitary Shimura varieties.
For the entire collection see [Zbl 1469.11002].Explicit equations for maximal curves as subcovers of the \(BM\) curvehttps://www.zbmath.org/1475.111262022-01-14T13:23:02.489162Z"Mendoza, Erik A. R."https://www.zbmath.org/authors/?q=ai:mendoza.erik-a-r"Quoos, Luciane"https://www.zbmath.org/authors/?q=ai:quoos.lucianeSummary: Let \(r\geq 3\) be an odd integer and \(\mathbb{F}_{q^{2r}}\) the finite field with \(q^{2r}\) elements. A second generalisation of the Giulietti-Korchmáros maximal curve over \(\mathbb{F}_{q^6}\) was presented in 2018 by \textit{P. Beelen} and \textit{M. Montanucci} [J. Lond. Math. Soc., II. Ser. 98, No. 3, 573--592 (2018; Zbl 1446.11119)], the so-called \(BM\) curve. This curve is maximal over \(\mathbb{F}_{q^{2r}}\) and isomorphic to the Giulietti-Korchmáros curve for \(r=3\). In this paper, benefiting from suitable representations of the automorphism group of the \(BM\) curve, we construct explicit equations for families of maximal algebraic curves as Galois subcovers of the \(BM\) curve, we also provide the genus and the Galois group associated to the subcover.On a generalization of Jacobi sumshttps://www.zbmath.org/1475.112162022-01-14T13:23:02.489162Z"Rojas-León, Antonio"https://www.zbmath.org/authors/?q=ai:rojas-leon.antonioSummary: We prove an estimate for multi-variable multiplicative character sums over affine subspaces of \(\mathbb{A}_k^n\), which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.Number of connected components of a real variety and \(\mathbb{R}\)-placeshttps://www.zbmath.org/1475.120022022-01-14T13:23:02.489162Z"Gondard-Cozette, Danielle"https://www.zbmath.org/authors/?q=ai:gondard-cozette.danielleSummary: The purpose of this paper is to present results and open problems related to \(\mathbb{R}\)-places. The first section recalls basic facts, the second introduces \(\mathbb{R}\)-places and their relationship with orderings and valuations.
The third part involves Real Algebraic Geometry and gives results proved using the space of \(\mathbb{R}\)-places. Theorem 14 gives explicitly, in terms of the function field of the variety, the number of connected components of a non-empty smooth projective real variety.
The fourth and fifth parts are devoted to the links with the real holomorphy rings and the valuation fans. Then we present an approach to abstract real places and conclude with some open questions.Algebraic groups as difference Galois groups of linear differential equationshttps://www.zbmath.org/1475.120112022-01-14T13:23:02.489162Z"Bachmayr, Annette"https://www.zbmath.org/authors/?q=ai:bachmayr.annette"Wibmer, Michael"https://www.zbmath.org/authors/?q=ai:wibmer.michaelLet \(F\) denote a \(\delta\sigma\)-field of characteristic zero, i.e. a field with two commuting operators, a derivation \(\delta\) and an endomorphism \(\sigma\). For such fields it is possible to build the theory of a similar Kolchin's Differential Galois theory. The authors in a number of publications [\textit{A. Bachmayr} et al., Doc. Math. 23, 241--291 (2018; Zbl 1436.12006); Adv. Math. 381, Article ID 107605, 28 p. (2021; Zbl 1461.12003); Trans. Am. Math. Soc. 374, No. 6, 4293--4308 (2021; Zbl 07344666)] make contribution to the development of such theory. This paper discusses the converse problem of the \(\sigma\)-Picard-Vessiot theory. In the usual Picard-Vessiot theory, for the field \(\mathbb{C}(x)\) and the linear algebraic group, the converse problem always has a solution [\textit{C. Tretkoff} and \textit{M. Tretkoff}, Am. J. Math. 101, 1327--1332 (1979; Zbl 0423.12021)]. The authors show that not every difference algebraic group occurs as a \(\sigma\)-Galois group of a \(\sigma\)-Picard-Vessiot extension of \(\mathbb{C}(x)\) . But the main result of the paper says that there are still quite a lot of positive occasions.
\textbf{Theorem.} \textit{Every linear algebraic group over \(\mathbb{C}\), considered as a difference algebraic group, occurs as a \(\sigma\)-Galois group over} \(\mathbb{C}(x)\).
Reviewer: Mykola Grygorenko (Kyïv)Effective difference elimination and nullstellensatzhttps://www.zbmath.org/1475.120122022-01-14T13:23:02.489162Z"Ovchinnikov, Alexey"https://www.zbmath.org/authors/?q=ai:ovchinnikov.alexey"Pogudin, Gleb"https://www.zbmath.org/authors/?q=ai:pogudin.gleb-a"Scanlon, Thomas"https://www.zbmath.org/authors/?q=ai:scanlon.thomas-jA sequence \((a_{j})_{j=0}^{\infty}\) of elements of a field \(K\) is said to be a solution of a difference equation with constant coefficients if there is a nonzero polynomial \(F(x_{0},\dots, x_{e})\in K[x_{0},\dots, x_{e}]\) such that for every natural number \(j\), one has \(F(a_{j}, a_{j+1},\dots, a_{j+e}) = 0\). This concept can be naturally generalized to systems of difference equations in several variables.
The paper under review answers the following fundamental questions about sequence solutions of systems of ordinary difference equations:
\begin{itemize}
\item[(i)] Under what conditions does such a system have a sequence solution?
\item[(ii)] Can these solutions be made sufficiently transparent to allow for efficient computation?
\item[(iii)] Given a system of difference equations on \((n+m)\)-tuples of sequences, how does one eliminate some of the variables so as to deduce the consequences of these equations on the first \(n\) variables?
\end{itemize}
As the answers to these questions, the authors prove two strong results the first of which (Theorem 3.1 of the paper) can be viewed as effective difference Nullstellensatz; it reduces the problem of solvability of a system of difference equations to the problem of consistency of certain system of finitely many polynomial equations. The second main result of the paper (Theorem 3.4) is an effective difference elimination theorem; it reduces the question of existing/finding a consequence in the \(\mathbf{x}\)-variables of a system of difference equations in \(\mathbf{x}\) and \(\mathbf{u}\) (\(\mathbf{x}=(x_{1},\dots, x_{m})\) and \(\mathbf{u}=(u_{1},\dots, u_{r})\) are two sets of variables) to a question about a polynomial ideal in a polynomial ring in finitely many variables.
Among other important results of the paper, one has to mention is a version of difference Nullstellensatz over an uncountable algebraically closed inversive difference field \(K\). It is shown that if \(F\) is a finite subset of the ring of difference polynomials \(K\{x_{1},\dots, x_{n}\}\), then the following statements are equivalent:
\begin{itemize}
\item[(i)] The system \(F=0\) has a solution in \(K^{\mathbb{Z}}\);
\item[(ii)] \(F=0\) has a solution in \(K^{\mathbb{N}}\);
\item[(iii)] \(F=0\) has finite partial solutions of length \(l\) for sufficiently large \(l\);
\item[(iv)] The difference ideal \(J\) generated by \(F\) in \(K\{x_{1},\dots, x_{n}\}\) does not contain \(1\);
\item[(v)] The reflexive closure of \(J\) in the inversive closure of \(K\{x_{1},\dots, x_{n}\}\) does not contain \(1\);
\item[(vi)] \(F=0\) has a solution in some difference \(K\)-algebra.
\end{itemize}
The paper also contains a number of examples that illustrate applications of the obtained results and counterexamples that show one cannot have a coefficient-independent effective strong Nullstellensatz for systems of difference equations.
Reviewer: Alexander B. Levin (Washington)Almost-simple affine difference algebraic groupshttps://www.zbmath.org/1475.120132022-01-14T13:23:02.489162Z"Wibmer, Michael"https://www.zbmath.org/authors/?q=ai:wibmer.michaelAffine algebraic groups can be described as subgroups of a general linear group defined by polynomials in the matrix entries. Affine difference algebraic groups can be described as subgroups of a general linear group defined by difference polynomials in the matrix entries, i.e., the defining equations involve a formal symbol \(\sigma\) that has to be interpreted as a ring endomorphism. The author shows that isomorphism theorems from abstract group theory have meaningful analogs for these groups and that one can establish a Jordan-Hölder type theorem that allows to decompose any affine difference algebraic group into almost-simple affine difference algebraic groups. The author also characterizes almost-simple affine difference algebraic groups via almost-simple affine algebraic groups.
Reviewer: Vladimir P. Kostov (Nice)Avoidance and absorbancehttps://www.zbmath.org/1475.130052022-01-14T13:23:02.489162Z"Tarizadeh, Abolfazl"https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazl"Chen, Justin"https://www.zbmath.org/authors/?q=ai:chen.justinIn this article, the authors investigate prime avoidance and prime absorbance by generalizing some classic results to radical ideals and infinite families of ideals in commutative rings with \(1 \neq 0\). We recall the prime avoidance lemma states that if an ideal is contained in a finite union of prime ideals, then it is already contained in one of them. The set-theoretic dual result, prime absorbance, if a finite intersection of ideals is contained in a prime ideal, then one of them is already contained in the prime. It is worth noting that both fail for infinite families. With this in mind, the authors prove several results about analogous notions involving radical ideals instead of prime ideals. The authors study compactly packed (C.P.) Rings where every set of primes has the avoidance property and rings in which every set of primes has the absorbance property, called properly zipped (or P.Z.) rings. The authors then provide several interesting characterizations of these rings and especially investigating the role chain conditions play. For example, it is shown that the only rings which are both C.P. and P.Z. rings are precisely the rings with finitely many prime ideals.
Reviewer: Christopher P. Mooney (Fulton)Quasi-cyclic modules and coregular sequenceshttps://www.zbmath.org/1475.130282022-01-14T13:23:02.489162Z"Hartshorne, Robin"https://www.zbmath.org/authors/?q=ai:hartshorne.robin"Polini, Claudia"https://www.zbmath.org/authors/?q=ai:polini.claudiaLet \(A\) be a ring, \(I\) an ideal, and \(M\) a non-zero \(A\)-module with \(\mathrm{Supp} (M)\subset V(I )\). A sequence \(x_1, \dots, x_r\) of elements of \(I\) is \textit{coregular for \(M\)} if \(x_1M = M\) and \(x_{i+1} (0 :_M (x_1,\dots, x_i )) = 0 :_M (x_1,\dots , x_i )\) for \(1 \leq i \leq r-1\). Recall that an irreducible curve in \(\mathbb{P}^3\) is called \textit{set-theoretic complete intersection} if it is the intersection of two surfaces.
\textit{M. Hellus} [Local cohomology and Matlis duality. Habilitationsschrift. Leipzig (2006)] gave the following criterion for a subvariety \(V\subset\mathbb{P}^n\) of codimension \(r\), with homogeneous ideal \(I\) in \(A\), the coordinate ring of \(\mathbb{P}^n\) to be a set-theoretic complete intersection: \(V\) is a set-theoretic complete intersection in \(\mathbb{P}^n\) if and only if the local cohomology modules \(H_I^i (A)\) are zero for all \(i\neq r\), and the Matlis dual \(D(H^r_I (A))\) has depth \(r\). In the paper under review, the authors develop the theory of coregular sequences and codepths and give a new version of Hellus's theorem: ``A variety \(V\) of codimension \(r\) in \(\mathbb{P}^n\) is a set-theoretic complete intersection if and only if \(H^i_I (A) = 0\) for all \(i > r\) and \(H^r_I (A)\) has codepth \(r\)'' (To avoid dealing with Matlis duals of large modules, they state their theorem in term of codepth). They also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. Then they focus on curves in \(\mathbb{P}^3\) and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in \(\mathbb{P}^3\) is a set-theoretic complete intersection.
Reviewer: Mohammad-Reza Doustimehr (Tabriz)On resurgence via asymptotic resurgencehttps://www.zbmath.org/1475.130352022-01-14T13:23:02.489162Z"DiPasquale, Michael"https://www.zbmath.org/authors/?q=ai:dipasquale.michael-r"Drabkin, Ben"https://www.zbmath.org/authors/?q=ai:drabkin.benLet \(I\) denote an ideal of the polynomial ring \(R\) over a field \(K\). For an integer \(s \geq 1\) let \(I^s\) resp \(I^{(s)}\) the \(s\)-th ordinary resp. the \(s\)-th symbolic power of \(I\). Clearly \(I^s \subseteq I^{(s)}\) for all \(s \geq 1\). In the paper [Math. Nachr. 129, 123--148 (1986; Zbl 0606.13001)] by the reviewer it is shown that for each \(r\) there is an integer \(s \geq r\) such that \(I^{(s)} \subseteq I^r\). The present author posed the problem to describe the smallest value \(f(r) := \min \{s \mid I^{(s)} \subseteq I^r\}\). Among others \textit{I. Swanson} [Math. Z. 234, No. 4, 755--775 (2000; Zbl 1010.13015)] proved that \(f(r)\) is bounded above by a linear function on \(r\). Under the above assumptions on \(R\) by the work of \textit{L. Ein} et al. [Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501)] it follows that \(f(r) \leq hr\), where \(h\) denotes the big height of \(I\). For more general rings further results are obtained by \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] and by \textit{L. Ma} and \textit{K. Schwede} [Invent. Math. 214, No. 2, 913--955 (2018; Zbl 1436.13009)]. A subtle point of view was initiated in the work of \textit{B. Harbourne} and \textit{C. Huneke} [J. Ramanujan Math. Soc. 28A, 247--266 (2013; Zbl 1296.13018)] with several research articles in recent times. This leads to two ``statistical quantities'', namely the resurgence \(\rho(I) = \sup \{s/r \mid I^{(s)} \not\subseteq I^r\}\) and the asymptotic resurgence \(\hat{\rho}(I) = s/r \mid I^{(st)} \not\subseteq I^{rt} \mbox{ for all } t \gg 0\}\). It follows that \(\hat{\rho}(I) \leq \rho(I) \leq h\). A problem is the rationality of the resurgence, answered in the affirmative by the authors provided the symbolic Rees algebra is finitely generated as an algebra. This is a consequence of the following: If \( \hat{\rho}(I) < \rho(I)\), then \( \rho(I)\) is a maximum instead of a supremum. This follows by the authors result about two bounds on asymptotic resurgence given a single known containment between a symbolic and ordinary power. With that they deduce subsequent criteria for expected resurgence, (that is, \( \rho(I) < h\)). As an application it is shown that squarefree monomial ideals have expected resurgence.
Reviewer: Peter Schenzel (Halle)\(\mathbb{A}_{\mathrm{inf}}\) is infinite dimensionalhttps://www.zbmath.org/1475.130372022-01-14T13:23:02.489162Z"Lang, Jaclyn"https://www.zbmath.org/authors/?q=ai:lang.jaclyn"Ludwig, Judith"https://www.zbmath.org/authors/?q=ai:ludwig.judithSummary: Given a perfect valuation ring \(R\) of characteristic \(p\) that is complete with respect to a rank-\(1\) nondiscrete valuation, we show that the ring \(\mathbb{A}_\mathrm{{\inf}}\) of Witt vectors of \(R\) has infinite Krull dimension.Doset Hibi rings with an application to invariant theoryhttps://www.zbmath.org/1475.130382022-01-14T13:23:02.489162Z"Miyazaki, Mitsuhiro"https://www.zbmath.org/authors/?q=ai:miyazaki.mitsuhiroAuthor's abstract: We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is \(F\)-rational otherwise. We also state criteria of Gorenstein property of these rings.
Reviewer: Kriti Goel (Mumbai)Infinite families of equivariantly formal toric orbifoldshttps://www.zbmath.org/1475.130392022-01-14T13:23:02.489162Z"Bahri, Anthony"https://www.zbmath.org/authors/?q=ai:bahri.anthony-p"Sarkar, Soumen"https://www.zbmath.org/authors/?q=ai:sarkar.soumen"Song, Jongbaek"https://www.zbmath.org/authors/?q=ai:song.jongbaekIn this paper, the analysis of the simplicial wedge construction on simplicial complexes and simple polytopes has been extended to the case of toric orbifolds. The authors present a class of infinite families of toric orbifolds which is derived from a given one by simplicial wedge construction. They show in their main result (Theorem 5.5) that its integral cohomology is free of torsion and is concentrated in even degrees.
Reviewer: Irem Portakal (Magdeburg)Multiplicity of the saturated special fiber ring of height three Gorenstein idealshttps://www.zbmath.org/1475.130492022-01-14T13:23:02.489162Z"Cid-Ruiz, Yairon"https://www.zbmath.org/authors/?q=ai:cid-ruiz.yairon"Mukundan, Vivek"https://www.zbmath.org/authors/?q=ai:mukundan.vivekSummary: Let \(R\) be a polynomial ring over a field and let \(I \subset R\) be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the \textit{saturated special fiber ring} of \(I\). The obtained formula depends only on the number of variables of \(R\), the minimal number of generators of \(I\), and the degree of the syzygies of \(I\). Applying results from \textit{L. Busé} et al. [Proc. Lond. Math. Soc. (3) 121, No. 4, 743--787 (2020; Zbl 1454.13017)] we get a formula for the \(j\)-multiplicity of \(I\) and an effective method to study a rational map determined by a minimal set of generators of \(I\).The Lefschetz theorem for hyperplane sectionshttps://www.zbmath.org/1475.140012022-01-14T13:23:02.489162Z"Hamm, Helmut A."https://www.zbmath.org/authors/?q=ai:hamm.helmut-a"Lê, Dũng Tráng"https://www.zbmath.org/authors/?q=ai:le-dung-trang.Summary: In these notes we consider different theorems of the Lefschetz type. We start with the classical Lefschetz Theorem for hyperplane sections on a non-singular projective variety. We show that this extends to the cases of a non-singular quasi-projective variety and to singular varieties. We also consider local forms of theorems of the Lefschetz type.
For the entire collection see [Zbl 1470.58001].Berkeley lectures on \(p\)-adic geometryhttps://www.zbmath.org/1475.140022022-01-14T13:23:02.489162Z"Scholze, Peter"https://www.zbmath.org/authors/?q=ai:scholze.peter"Weinstein, Jared"https://www.zbmath.org/authors/?q=ai:weinstein.jaredThe following review consists partly of quotations (and points even slightly altered) from the book under discussion, partly of extracts formulated by the reviewer; the distinction is not made explicit.\\
\textbf{From the Foreword:}
This is a revised version of the lecture notes for the course on \(p\)-adic geometry given by Peter Scholze in the fall of 2014 at UC Berkeley.
In the first half of the course (Lectures 1--10) we construct the category of diamonds, which are quotients of perfectoid spaces by so-called pro-étale equivalence relations. In brief, diamonds are to perfectoid spaces what algebraic spaces are to schemes.
In the second half of the course (Lectures 11--25), we define spaces of mixed-characteristic local shtukas, which live in the category of diamonds. This requires making sense of products like \(\mathrm{Spa}{\mathbb Q}_p \times S\), where \(S\) is an adic space over \({\mathbb F}_p\).
The proper foundations on diamonds can only be found in [Peter Scholze: Étale cohomology of diamonds, 2017]; here, we only survey the main ideas in the same way as in the original lectures. In this way, we hope that this manuscript can serve as an informal introduction to these ideas.
\textbf{From the Introduction:}
1.1 Motivation: Drinfeld, L. Lafforgue and V. Lafforgue
The starting point is Drinfeld's work on the global Langlands correspondence over function fields. Fix \(X/{\mathbb F}_p\) a smooth projective curve, with function field \(K\). The Langlands correspondence for \(\mathrm{GL}_n /K\) is a bijection \(\pi\mapsto\sigma(\pi)\) between the following two sets (considered up to isomorphism):
\({\bullet}\) Cuspidal automorphic representations of \(\mathrm{GL}_n(\mathbf{A}_K)\), where \(\mathbf{A}_K\) is the ring of adeles of \(K\), and
\({\bullet}\) Irreducible representations \(\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}_n(\overline{\mathbb Q}_{\ell})\).
Whereas the global Langlands correspondence is largely open in the case of number fields \(K\), it is a theorem for function fields, due to Drinfeld (\(n = 2\)) and L. Lafforgue (general \(n\)). The key innovation in this case is Drinfeld's notion of an \(X\)-shtuka.
A family of stacks \[f:\mathrm{Sht}_{\mathrm{GL}_n,\{\mu_1,\ldots,\mu_m\},N}\longrightarrow (X\backslash N)^m\](depending on a divisor \(N\) on \(X\) and on additional data \(\mu_1,\ldots,\mu_m\)) is considered, and then the cohomology \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\), where \(d\) is the relative dimension of \(f\).
The analysis of \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) runs along the following lines, explained by means of a \(\overline{\mathbb Q}_{\ell}\)-sheaf \({\mathbb L}\) on \(X^m\) which becomes lisse when restricted to \(U^m\) for some dense open subset \(U\) of \(X\). We can think of \({\mathbb L}\) as a representation of the étale fundamental group \(\pi_1(U^m)\) on a \(\overline{\mathbb Q}_{\ell}\)-vector space. Ultimately we want to relate this to \(\pi_1(U)\), because this is a quotient of \(\mathrm{Gal}(\overline{K}/K)\).
For \(i =1,\ldots, m\) we have a partial Frobenius map \(F_i: X^m\to X^m\), which is \(\mathrm{Frob}_X\) on the \(i\)-th factor, and the identity on each other factor. For an étale morphism \(V\to X^m\), let us say that a system of partial Frobenii on \(V\) is a commuting collection of isomorphisms \(F_i^*V \cong V\) over \(X^m\) (and whose product is the relative Frobenius of \(V\to X^m\)). Finite étale covers of \(U^m\) equipped with partial Frobenii form a Galois category, and thus they are classified by continuous actions of a profinite group \(\pi_1(U^m/\mathrm{ partial Frob})\) on a finite set.\\
\textbf{Lemma:} (Drinfeld) The natural map\[\pi_1(U^m/\mathrm{partial Frob})\longrightarrow \longrightarrow\pi_1(U)\times\cdots\times\pi_1(U)\quad\quad(m\,\, \mathrm{copies})\] is an isomorphism.
Now \(R^d(f_N)_!\overline{\mathbb Q}_{\ell}\) is not exactly like an \({\mathbb L}\) as above, but only approximately so, yet the Lemma can be suitably adapted so that one gets a big representation of \[\mathrm{GL}_n(\mathrm{A}_K)\times\mathrm{ Gal}(\overline{K}/K)\times\cdots\times\mathrm{Gal}(\overline{K}/K)\] on \(\mathrm{lim}_{\to N}R^d(f_N)_!\overline{\mathbb Q}_{\ell}\). One then expects the latter to decompose under the said action to yield the desired Langlands correspondence.\\
1.2 The possibility of Shtukas in mixed characteristic
It would be desirable to have moduli spaces of shtukas over number fields, but the first immediate problem is that such a space of shtukas would live over something like \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\), where the product is over \({\mathbb F}_1\) somehow.
In this course we will give a rigorous definition of \(\mathrm{Spec}({\mathbb Z}_p)\times\mathrm{Spec}({\mathbb Z}_p)\), the completion of \(\mathrm{Spec}({\mathbb Z})\times\mathrm{Spec}({\mathbb Z})\) at \((p, p)\). It lives in the world of nonarchimedean analytic geometry, so it should properly be called \(\mathrm{Spa}({\mathbb Z}_p)\times\mathrm{Spa}({\mathbb Z}_p)\). (The notation \(\mathrm{Spa}\) refers to the adic spectrum.)
Whatever it is, it should contain \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\) as a dense open subset.
We present a model for the product \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), specified by picking one of the factors: one copy of \({\mathbb Q}_p\) appears as the field of scalars, but the other copy appears geometrically. Consider the open unit disc \(D_{{\mathbb Q}_p} = \{x\,;\, |x| < 1\}\) as a subgroup of (the adic version of) \({\mathbb G}_m\), via \(x\mapsto 1 + x\). Then \(D_{{\mathbb Q}_p}\) is in fact a \({\mathbb Z}_p\)-module with multiplication by \(p\) given by \(x\mapsto (1 + x)^p-1\), and we consider\[\widetilde{D}_{{\mathbb Q}_p}=\projlim_{x\mapsto (1 + x)^p-1} D_{{\mathbb Q}_p}.\] After base extension to a perfectoid field, this is a perfectoid space, which carries the structure of a \({\mathbb Q}_p\)-vector space. Thus its punctured version \(\widetilde{D}^*_{{\mathbb Q}_p}=\widetilde{D}_{{\mathbb Q}_p}-\{0\}\) has an action of \({\mathbb Q}_p^{\times}\), and we consider the quotient \[\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p):=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}.\]Note that this quotient does not exist in the category of adic spaces (the quotient being taken in a formal sense). On \(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times}\), we have an operator \(\varphi\), corresponding to \(p\in{\mathbb Q}_p^{\times}\). Let \[X =(\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Z}_p^{\times})/\varphi^{\mathbb Z}=\widetilde{D}^*_{{\mathbb Q}_p}/ {\mathbb Q}_p^{\times}.\]One can define a finite étale cover of \(X\) simply as a \({\mathbb Q}_p^{\times}\)-equivariant finite étale cover of \(\widetilde{D}^*_{{\mathbb Q}_p}\). There is a corresponding profinite group \(\pi_1(X)\) which classifies such covers. We have the following theorem, which is a local version of Drinfeld's lemma in the case \(m = 2\).\\
\textbf{Theorem:} \[\pi_1(X) \cong \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p).\]
This theorem suggests that if one could define a moduli space of \({\mathbb Q}_p\)-shtukas which is fibered over products such as \(\mathrm{Spa}({\mathbb Q}_p)\times\mathrm{Spa}({\mathbb Q}_p)\), then its cohomology would produce representations of \(\mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\times \mathrm{Gal}(\overline{\mathbb Q}_p/{\mathbb Q}_p)\).
What would a \({\mathbb Q}_p\)-shtuka over \(S\) look like? It should be a vector bundle \({\mathcal E}\) over \(\mathrm{Spa} {\mathbb Q}_p\times S\), together with a meromorphic isomorphism \(\mathrm{Frob}^*_S{\mathcal E}- - \to{\mathcal E}\). In order for this to make any sense, we would need to give a geometric meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) (and to \(\mathrm{Frob}_S\)) just as we gave one to \(\mathrm{Spa}({\mathbb Q}_p)\times \mathrm{Spa}({\mathbb Q}_p)\).
We will give a meaning to \(\mathrm{Spa} {\mathbb Q}_p\times S\) whenever \(S\) is a perfectoid space of characteristic \(p\), which lets us define moduli spaces of \(p\)-adic shtukas. In general, these are not representable by perfectoid spaces or classical rigid spaces, but instead they are diamonds: That is, quotients of perfectoid spaces by pro-étale equivalence relations. A large part of this course is about the definition of perfectoid spaces and diamonds.\\
\textbf{Table of contents:}
Lecture 1: Introduction
Lecture 2: Adic spaces
Lecture 3: Adic spaces II
Lecture 4: Examples of adic spaces
Lecture 5: Complements on adic spaces
Lecture 6: Perfectoid rings
Lecture 7: Perfectoid spaces
Lecture 8: Diamonds
Lecture 9: Diamonds II
Lecture 10: Diamonds associated with adic spaces
Lecture 11: Mixed characteristic shtukas
Lecture 12: Shtukas with one leg
Lecture 13: Shtukas with one leg II
Lecture 14: Shtukas with one leg III
Lecture 15: Examples of diamonds
Lecture 16: Drinfel's lemma for diamonds
Lecture 17: The v-topolgy
Lecture 18: v-sheaves associated with perfect and formal schemes
Lecture 19: The \(B_{dR}^+\)-affine Grassmannian
Lecture 20: Families of affine Grassmannians
Lecture 21: Affine flag varieties
Lecture 22: Vector bundles and \(G\)-torsors
Lecture 23: Moduli spaces of shtukas
Lecture 24: Local Shimura varieties
Lecture 25: Integral models of local Shimura varieties
Reviewer: Elmar Große-Klönne (Berlin)Moduli spaces and locally symmetric spaces. Based on two workshops, Morningside Center of Mathematics, Beijing, China, February 2017 and March 2019https://www.zbmath.org/1475.140032022-01-14T13:23:02.489162ZPublisher's description: This book consists of five expository papers on moduli spaces and locally symmetric spaces based on lecture notes given by the authors at two instructional workshops held at the Morningside Center of Mathematics, Beijing, in February 2017 and March 2019. They give accessible and systematic introductions to moduli spaces of Riemann surfaces, algebraic curves, moduli spaces of vector bundles on Riemann surfaces, moduli spaces of singularities, and compactification of a natural class of locally symmetric spaces. They should serve as good instructions to some important aspects of these important spaces.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Gasbarri, Carlo}, Diophantine geometry on curves over function fields, 1-38 [Zbl 1475.14048]
\textit{Hertling, Claus; Roucairol, Céline}, Distinguished bases and Stokes gegions for the simple and the simple elliptic singularities, 39-106 [Zbl 07442148]
\textit{Kasparian, A. K.; Sankaran, G. K.}, Toroidal compactification: the generalised ball case, 107-133 [Zbl 1475.14099]
\textit{Papadopoulos, Athanase}, Ideal triangles, hyperbolic surfaces and the Thurston metric on Teichmüller space, 135-181 [Zbl 07442150]
\textit{Schmitt, Alexander H. W.}, Vector bundles over compact Riemann surfaces, 183-359 [Zbl 07442151]Structural results on harmonic rings and lessened ringshttps://www.zbmath.org/1475.140042022-01-14T13:23:02.489162Z"Tarizadeh, Abolfazl"https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazl"Aghajani, Mohsen"https://www.zbmath.org/authors/?q=ai:aghajani.mohsenSummary: In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring \(A\) modulo its Jacobson radical is a zero dimensional ring, then \(A\) is a clean ring. It is also proved that, for a given Gelfand ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Max}(A)\) is flat continuous if and only if \(A\) modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Min}(A)\) is Zariski continuous if and only if \(\mathrm{Min}(A)\) is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes ``reduced ring'' notion. Especially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring \(A\) is a finite product of lessened quasi-prime rings if and only if \(\mathrm{Ker}\pi_{\mathfrak{p}}\) is a finitely generated and idempotent ideal for all \(\mathfrak{p}\in\mathrm{Min}(A)\).A geometric invariant of \(6\)-dimensional subspaces of \(4\times 4\) matriceshttps://www.zbmath.org/1475.140052022-01-14T13:23:02.489162Z"Chirvasitu, Alex"https://www.zbmath.org/authors/?q=ai:chirvasitu.alexandru"Smith, S. Paul"https://www.zbmath.org/authors/?q=ai:smith.s-paul"Vancliff, Michaela"https://www.zbmath.org/authors/?q=ai:vancliff.michaelaSummary: Let \(R\) denote a 6-dimensional subspace of the ring \(M_4(\Bbbk )\) of \(4 \times 4\) matrices over an algebraically closed field \(\Bbbk \). Fix a vector space isomorphism \(M_4(\Bbbk ) \cong \Bbbk^4 \otimes \Bbbk^4\). We associate to \(R\) a closed subscheme \({\mathbf X}_R\) of the Grassmannian of 2-dimensional subspaces of \(\Bbbk^4\), where the reduced subscheme of \({\mathbf X}_R\) is the set of 2-dimensional subspaces \(Q \subseteq \Bbbk^4\) such that \((Q \otimes \Bbbk^4) \cap R \ne \{ 0\}\). Our main result is that if \({\mathbf X}_R\) has minimal dimension (namely, one), then its degree is 20 when it is viewed as a subscheme of \(\mathbb{P}^5\) via the Plücker embedding.
We present several examples of \(\mathbf X_R\) that illustrate the wide range of possibilities for it; there are reduced and non-reduced examples. Two examples involve elliptic curves: in one case, \( {\mathbf X}_R\) is a \(\mathbb{P}^1\)-bundle over an elliptic curve the second symmetric power of the curve; in the other, it is a curve having seven irreducible components, three of which are quartic elliptic space curves, and four of which are smooth plane conics. These two examples arise naturally from a problem having its roots in quantum statistical mechanics.
The scheme \(\mathbf X_R\) appears in non-commutative algebraic geometry: under appropriate hypotheses, it is isomorphic to the line scheme \(\mathcal{L}\) of a certain graded algebra determined by \(R\). In that context, it has been an open question for several years to describe such \(\mathcal{L}\) of minimal dimension, i.e., those \(\mathcal{L}\) of dimension one. Our main result implies that if \(\dim (\mathcal{L}) = 1\), then, as a subscheme of \(\mathbb{P}^5\) under the Plücker embedding, \( \deg (\mathcal{L}) = 20\).\(K3\) carpets on minimal rational surfaces and their smoothingshttps://www.zbmath.org/1475.140062022-01-14T13:23:02.489162Z"Bangere, Purnaprajna"https://www.zbmath.org/authors/?q=ai:bangere.purnaprajna"Mukherjee, Jayan"https://www.zbmath.org/authors/?q=ai:mukherjee.jayan"Raychaudhury, Debaditya"https://www.zbmath.org/authors/?q=ai:raychaudhury.debadityaAll the algebraic schemes in this paper are over the field \(\mathbb C\). One studies (non-split abstract) double structures on \({\mathbb P}^2\) and on Hirzebruch surfaces \({\mathbb F}_e= \hbox{projective bundle over } {\mathbb P}^1\) associated to the vector bundle \({\mathcal O}_{{\mathcal P}^1} \oplus {\mathcal O}_{{\mathcal P}^1}(-e) \), which have trivial canonical sheaf and are regular; such structures are called \(K3\) {\em carpets}. One shows in Theorem 3.1 that, in the case of \({\mathbb F }_e\), these are parametrised by the projective line, in general are not projective and those which are projective are parametrized by a countable set. For \({\mathbb P}^2\) the situation is different, by results of \textit{J.-M. Drézet} [``Primitive multiple schemes'', Preprint, \url{arXiv:2004.04921}], who proved the existence and the unicity of a non-split abstract \(K3\) structure.
The paragraph 4 deals with ``Smoothings of Abstract and Embedded \(K3\) Carpets''. One shows ``that all projective \(K3\) carpets can be smoothed to a smooth \(K3\) surface'', separately for support \({\mathbf F }_e\) with \(0\le e \le 2\) and \(e \ge 3\), because the treatment is different in the two cases: for the first case there exist smooth \(K3\) double covers \(X \rightarrow {\mathbf F }_e\), for the second one uses a degeneration argument.
In the last part of the paper one studies embedded \(K3\) carpets, in particular one shows (Theorem 5.1) that an embedded \(K3\) carpet with support a Hirzebruch surface \({\mathbb F }_e\) is represented by a smooth point in the corresonding Hilbert scheme iff \(0\le e \le 2\).
Reviewer: Nicolae Manolache (Bucureşti)Chow group of 1-cycles of the moduli of parabolic bundles over a curvehttps://www.zbmath.org/1475.140072022-01-14T13:23:02.489162Z"Chakraborty, Sujoy"https://www.zbmath.org/authors/?q=ai:chakraborty.sujoyGiven a nonsingular projective curve \(X\) of genus \(g \ge 3\) over \(\mathbb{C}\), let \(\mathcal{M}(r, \mathcal{L})\) denote the moduli space of stable bundles over \(X\) of rank \(r\) with fixed determinant \(\mathcal{L}\). Moreover let \(\mathcal{M}(r, \mathcal{L}, \alpha )\) denote the moduli space of parabolic bundles of full flags along the fixed parabolic points and generic weights \(\alpha\). The author proves the following result on the Chow group of 1-cycles with rational coefficient:
Theorem 1.1. (Theorem 3.12) For any two generic weights \(\alpha\) and \(\beta\), there exists a canonical isomorphism
\[
\mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\alpha) \cong \mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\beta)
\]
On the other hand for \(\mathcal{L} = \mathcal{O}_X(x)\), it has been shown in [\textit{I. Choe} and \textit{J.-M. Hwang}, Math. Z. 253, No. 2, 281--293 (2006; Zbl 1102.14002)] that
\[
\mathrm{CH}_1^{\mathbb{Q}} ( \mathcal{M} (2, \mathcal{L}) \cong \mathrm{CH}_0^{\mathbb{Q}} (X). \tag{1}
\]
Combining with this result, the author shows:
Theorem 1.2 (Theorem 4.6) In case of rank 2 and determinant \(\mathcal{O}_X(x)\), for any generic weight \(\alpha\), we have
\[
\mathrm{CH}_1^{\mathbb{Q}} (\mathcal{M}_\alpha) \cong \mathbb{Q}^n \oplus \mathrm{CH}_0^{\mathbb{Q}} (X),
\]
where \(n\) is the number of points of the parabolic data.
Main idea is to show that for a sufficiently small generic \(\alpha\), the space \(\mathcal{M}_\alpha\) has a structure of a \((\mathbb{P}^1)^n\)-bundle over \(\mathcal{M}(2, \mathcal{O}_X(x))\).
Remark: Recently the isomorphism (1) has been improved to an isomorphism with integer coefficient by D. Li, Y. Lin and X. Pan [\textit{D. Li} et al., C. R., Math., Acad. Sci. Paris 357, No. 2, 209--211 (2019; Zbl 1408.14043)]. It seems that Theorem 1.2 above can also be improved in view of this result.
Reviewer: Insong Choe (Seoul)Multiplicative Chow-Künneth decompositions and varieties of cohomological \(K3\) typehttps://www.zbmath.org/1475.140082022-01-14T13:23:02.489162Z"Fu, Lie"https://www.zbmath.org/authors/?q=ai:fu.lie"Laterveer, Robert"https://www.zbmath.org/authors/?q=ai:laterveer.robert"Vial, Charles"https://www.zbmath.org/authors/?q=ai:vial.charlesA Chow-Künneth decomposition of a smooth projective variety \(X\) is a direct-sum decomposition of its rational Chow motive. This in particular says that the diagonal in \(X\times X\) can be written as a sum of orthogonal projectors to the graded parts of the Chow ring. By Murre's conjecture, every smooth projective variety is supposed to have a Chow-Künneth decomposition. There is a natural notion of multiplicativity of a Chow-Künneth decomposition and admitting a multiplicative Chow-Künneth decomposition (MCK) is a cycle theoretic property. The question arises which varieties admit an MCK decomposition and the article under review contributes to it.
\textit{M. Shen} and \textit{C. Vial} conjectured in [The Fourier transform for certain hyperkähler fourfolds. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1386.14025)] that hyperkähler varieties admit an MCK. On the other hand, it is known that Fano varieties and canonically polarized varieties do not in general admit an MCK. Examples can be found in the present article (Example 2.11 due to Beauville and Example 3.3 due to Fu-Vial and Ceresa). The question thus becomes to find classes of varieties with ample or anti-ample canonical sheaf that do admit an MCK decomposition. In the present article, the authors collect evidence that on the Fano side, varieties which are cohomologically of \(K3\) type have an MCK decomposition. This is proven for cubic fourfolds (in fact, for cubic hypersurfaces in general, see Corollary 1.3 and Theorem 5.4) and for Küchle fourfolds of type c7 (Theorem 6.2). Also for canonically polarized varieties, the authors establish the existence of an MCK decomposition for two families of Todorov surfaces (Theorem~1.4). Contrary to previously known examples, these surfaces are not birational to products of curves.
Reviewer: Christian Lehn (Chemnitz)Some torsion classes in the Chow ring and cohomology of \(\mathbf{B}PGL_n\)https://www.zbmath.org/1475.140092022-01-14T13:23:02.489162Z"Gu, Xing"https://www.zbmath.org/authors/?q=ai:gu.xingSummary: In the integral cohomology ring of the classifying space of the projective linear group \(\mathrm{PGL}_n\) (over \(\mathbb{C} )\), we find a collection of \(p\)-torsion classes \(y_{p , k}\) of degree \(2 ( p^{k + 1} + 1 )\) for any odd prime divisor \(p\) of \(n\), and \(k \geqslant 0\). If, in addition, \( p^2 \nmid n\), there are \(p\)-torsion classes \(\rho_{p , k}\) of degree \(p^{k + 1} + 1\) in the Chow ring of the classifying stack of \(\mathrm{PGL}_n\), such that the cycle class map takes \(\rho_{p , k}\) to \(y_{p , k}\). We present an application of the above classes regarding Chern subrings.On the motive of Kapustka-Rampazzo's Calabi-Yau threefoldshttps://www.zbmath.org/1475.140102022-01-14T13:23:02.489162Z"Laterveer, Robert"https://www.zbmath.org/authors/?q=ai:laterveer.robertSummary: In [Commun. Number Theory Phys. 13, No. 4, 725--761 (2019; Zbl 1451.14024)], \textit{M. Kapustka} and \textit{M. Rampazzo} have exhibited pairs of Calabi-Yau threefolds \(X\) and \(Y\) that are L-equivalent and derived equivalent, without being birational. We complete the picture by showing that \(X\) and \(Y\) have isomorphic Chow motives.Rost motives, affine varieties, and classifying spaceshttps://www.zbmath.org/1475.140112022-01-14T13:23:02.489162Z"Petrov, Victor"https://www.zbmath.org/authors/?q=ai:petrov.viktor"Semenov, Nikita"https://www.zbmath.org/authors/?q=ai:semenov.nikitaSummary: In this article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety of full flags of a split semisimple algebraic group \(G\) over a field, a motivic decomposition of \(E/B\) for a \(G\)-torsor \(E\) over a smooth base scheme, study torsion subgroup of the Chow group of \(E/B\), define some equivariant Rost motives over a field and some ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.Hopf-theoretic approach to motives of twisted flag varietieshttps://www.zbmath.org/1475.140122022-01-14T13:23:02.489162Z"Petrov, Victor"https://www.zbmath.org/authors/?q=ai:petrov.viktor"Semenov, Nikita"https://www.zbmath.org/authors/?q=ai:semenov.nikitaSummary: Let \(G\) be a split semisimple algebraic group over a field and let \(A^*\) be an oriented cohomology theory in the Levine-Morel sense. We provide a uniform approach to the \(A^*\)-motives of geometrically cellular smooth projective \(G\)-varieties based on the Hopf algebra structure of \(A^*(G)\). Using this approach, we provide various applications to the structure of motives of twisted flag varieties.Dedekind sums and parsing of Hilbert serieshttps://www.zbmath.org/1475.140132022-01-14T13:23:02.489162Z"Zhou, Shengtian"https://www.zbmath.org/authors/?q=ai:zhou.shengtianSummary: Given a polarized variety \(X,D\), we can associate a graded ring and a Hilbert series. Assume \(D\) is an ample \(\mathbb{Q}\) Cartier divisor, and \(X,D\) is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension \(\le 1\), that is, both singularities of dimension \(1\) and singular points, extending a result of \textit{A. Buckley} et al. [Izv. Math. 77, No. 3, 461--486 (2013; Zbl 1273.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 3, 29--54 (2013)] about varieties with only isolated singularities.Fibration of \(\log \)-general type space over quasi-abelian varietieshttps://www.zbmath.org/1475.140142022-01-14T13:23:02.489162Z"Wei, Chuanhao"https://www.zbmath.org/authors/?q=ai:wei.chuanhaoSummary: We show that there exists no smooth fibration of a smooth complex quasi-projective variety of \(\log \)-general type over a quasi-abelian variety. The proof uses \textit{M. Popa} and \textit{C. Schnell}'s [Ann. Math. (2) 179, No. 3, 1109--1120 (2014; Zbl 1297.14011)] construction of Higgs bundle.Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensionshttps://www.zbmath.org/1475.140152022-01-14T13:23:02.489162Z"Gurjar, Sudarshan"https://www.zbmath.org/authors/?q=ai:gurjar.sudarshan-rajendra"Nitsure, Nitin"https://www.zbmath.org/authors/?q=ai:nitsure.nitinSummary: Let \(G\) be a connected split reductive group over a field \(k\) of characteristic zero. Let \(X\rightarrow S\) be a smooth projective morphism of \(k\)-schemes, with geometrically connected fibers. We formulate a natural definition of a relative canonical reduction, under which principal \(G\)-bundles of any given Harder-Narasimhan type \(\tau \) on fibers of \(X/S\) form an Artin algebraic stack \(Bun_{X/S}^{\tau }(G)\) over \(S\), and as \(\tau \) varies, these stacks define a stratification of the stack \(Bun_{X/S}(G)\) by locally closed substacks. This result extends to principal bundles in higher dimensions the earlier such result for principal bundles on families of curves. The result is new even for vector bundles, that is, for \(G = GL_{n,k}\).Instanton bundles on the Segre threefold with Picard number threehttps://www.zbmath.org/1475.140162022-01-14T13:23:02.489162Z"Antonelli, V."https://www.zbmath.org/authors/?q=ai:antonelli.vincenzo"Malaspina, F."https://www.zbmath.org/authors/?q=ai:malaspina.francescoThe authors provide a description of \(\mu\)-semistable instanton bundles (of charge \(k\)) on \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\) in terms of cohomology of monads. The proof is based on a stronger version of the Beilinson spectral sequence (on \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\)) and a detailed analysis of full exceptional collections. A characterization of Gieseker strictly semistable Instanton bundles in terms of cohomology of monads is also proved.
In the second part of the paper, the authors prove the existence of \(\mu\)-semistable instanton bundles on \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\) for every charge \(k\). The proof is based on induction: first to construct a torsion free sheaf with larger \(c_2\) and then deform it to a vector bundle. As a consequence, one gets the existence of a generically smooth irreducible component of the module space of Instanton bundles (with appropriate numerical data).
At the end, a description of jumping lines of generic Instanton bundles on \(\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1\), contained in the rulings, is given.
Reviewer: Krishanu Dan (Odisha)Monodromy of rank 2 parabolic Hitchin systemshttps://www.zbmath.org/1475.140172022-01-14T13:23:02.489162Z"Kydonakis, Georgios"https://www.zbmath.org/authors/?q=ai:kydonakis.georgios"Sun, Hao"https://www.zbmath.org/authors/?q=ai:sun.hao"Zhao, Lutian"https://www.zbmath.org/authors/?q=ai:zhao.lutianThe aim of this paper is to study the monodromy of the Hitchin fibration for moduli spaces of parabolic \(G\)-Higgs bundles in the cases when \(G=SL(2,\mathbb{R})\), \(GL(2,\mathbb{R})\) and \(PGL(2,\mathbb{R})\). A calculation of the orbits of the monodromy with \(\mathbb{Z}_2\)-coefficients provides an exact count of the components of the moduli spaces for these groups. This paper is organized as follows : The first section is an introduction to the subject and summarizes the main results. In the second section the authors introduce terminology for the moduli spaces of parabolic \(G\)-Higgs bundles that they are primarily interested in this paper. In section 3, they consider the Hitchin fibration and the construction of the spectral curve for rank \(2\) parabolic Hitchin systems. They also discuss here the parabolic version of the BNR correspondence with particular focus on the subvarieties of the Picard group restricted to which the correspondence is one-to-one, as well as on the Prym variety of the spectral covering of the \(V\)-surface. Moduli spaces of parabolic (or non-parabolic) \(G\)-Higgs bundles can be decomposed into closed subvarieties, yet not necessarily connected components, for fixed values of appropriate topological invariants. In this section the authors describe such topological invariants for the moduli spaces they are interested in, namely for the rank 2 cases \(G=SL(2,\mathbb{R})\), \(GL(2,\mathbb{R})\) and \(PGL(2,\mathbb{R})\). Here, the authors include the discussion for the topological invariants leading to the minimum number of connected components, while in section 5 they study the monodromy action on \(2\)-torsion points on the lattices. Finally, section 6 and section 7 include the exact calculation of the number of orbits of the monodromy action. The paper is supported by an appendix concerning BNR correspondence for orbicurves.
Reviewer: Ahmed Lesfari (El Jadida)Properties of schemes of morphisms and applications to blow-upshttps://www.zbmath.org/1475.140182022-01-14T13:23:02.489162Z"das Dores, Lucas"https://www.zbmath.org/authors/?q=ai:das-dores.lucasSummary: Let \(X\) be a fixed projective scheme which is flat over a base scheme \(S\). The association taking a quasi-projective \(S\)-scheme \(Y\) to the scheme parametrizing \(S\)-morphisms from \(X\) to \(Y\) is functorial. We prove that this functor preserves limits, and both open and closed immersions. As an application, we determine a partition of schemes parametrizing rational curves on the blow-ups of projective spaces at finitely many points. We compute the dimensions of its components containing rational curves outside the exceptional divisor and the ones strictly contained in it. Furthermore, we provide an upper bound for the dimension of the irreducible components intersecting the exceptional divisors properly.A new method toward the Landau-Ginzburg/Calabi-Yau correspondence via quasi-mapshttps://www.zbmath.org/1475.140192022-01-14T13:23:02.489162Z"Choi, Jinwon"https://www.zbmath.org/authors/?q=ai:choi.jinwon"Kiem, Young-Hoon"https://www.zbmath.org/authors/?q=ai:kiem.young-hoonSummary: The Landau-Ginzburg/Calabi-Yau correspondence claims that the Gromov-Witten invariant of the quintic Calabi-Yau 3-fold should be related to the Fan-Jarvis-Ruan-Witten invariant of the associated Landau-Ginzburg model via wall crossings. In this paper, we consider the stack of quasi-maps with a cosection and introduce sequences of stability conditions which enable us to interpolate between the moduli stack for Gromov-Witten invariants and the moduli stack for Fan-Jarvis-Ruan-Witten invariants.Betti spectral gluinghttps://www.zbmath.org/1475.140202022-01-14T13:23:02.489162Z"Ben-Zvi, David"https://www.zbmath.org/authors/?q=ai:ben-zvi.david"Nadler, David"https://www.zbmath.org/authors/?q=ai:nadler.david\textit{D. Arinkin} and \textit{D. Gaitsgory} [Sel. Math., New Ser. 21, No. 1, 1--199 (2015; Zbl 1423.14085)]
showed in the de Rham setting of the geometric Langlands correspondence that one needs to shift from quasi-coherent sheaves to ind-coherent sheaves with nilpotent singular support on the spectral side to preserve parabolic induction. In prior works, the authors proposed a similar shift from perfect complexes to coherent sheaves in the Hochshild setting, and developed techniques for working with coherent sheaves, such as descent with prescribed singular support. The main result of this paper applies these techniques to show that the proposed Betti spectral category enjoys the gluing properties expected from TFT.
More precisely, let \(S\) be a surface, \(G\) a complex reductive group, and \(\mathcal{L}oc_G(S)\) the moduli of \(G\)-local systems on \(S\) considered as a derived stack. Then the Betti spectral category is the dg category of coherent sheaves with nilpotent singular support \(\textrm{DCoh}_{\mathcal{N}}(\mathcal{L}oc_G(S))\). Further, let \(B\subset G\) be a Borel subgroup and let \(\mathcal{L}oc_G(S,\partial S)\) be the parabolic derived stack of \(G\)-local systems on \(S\) with \(B\)-reductions along \(\partial S\). In the case of the cylinder \(S=S^1\times[0,1]\) it is the Grothendieck-Steinberg stack \(St_G\), and \(\mathcal{H}_G:=\textrm{DCoh}(St_G)\) is the spectral affine Hecke category equipped with monoidal structure induced by concatenation. Identifying two boundary components of \(\partial S\) with the same circle produces both a new surface \(\widetilde{S}\) and a natural bimodule structure on \(\textrm{DCoh}_{\mathcal{N}}(\mathcal{L}oc_G(S,\partial S))\). The main result states that there is a canoncial equivalence \[ \textrm{DCoh}_{\mathcal{N}}(\mathcal{L}oc_G(\widetilde{S},\partial\widetilde{S}))\simeq\mathcal{H}_G\otimes_{\mathcal{H}_G}\otimes_{\mathcal{H}_G^{op}}\textrm{DCoh}_{\mathcal{N}}(\mathcal{L}oc_G(S,\partial S)) \] respecting commuting Wilson line operators realized by Hecke modifications at points (the action of \(\textrm{Perf}(\mathcal{L}oc_G(S))\)\,) and Verlinde loop operators realized by Hecke modifications along closed loops (the action of the center of \(\mathcal{H}_G\)). The authors call it the spectral Verlinde formula by analogy to braided tensor categories in TFT, and it allows one to reduce the description of the categories attached to arbitrary surfaces to those of disks, cylinders, pairs of pants, and Möbius bands.
The result is deduced from a more abstract version for smooth derived stacks proved by descent with singular support, which also allows to specify arbitrary ramification conditions for \(G\)-local systems on \(S\) in terms of \(\mathcal{H}_G\) modules. These results suggest existence of a fully extended \((3 + 1)\)-dimensional TFT that assigns \(\textrm{DCoh}_{\mathcal{N}}(\mathcal{L}oc_G(S))\) to a surface \(S\), and the 2-category of small \(\mathcal{H}_G\)-module categories to the circle \(S^1\). A natural next step is to identify a suitable 3-category to assign to the point.
Reviewer: Sergiy Koshkin (Houston)An analytic version of the Langlands correspondence for complex curveshttps://www.zbmath.org/1475.140212022-01-14T13:23:02.489162Z"Etingof, Pavel"https://www.zbmath.org/authors/?q=ai:etingof.pavel-i"Frenkel, Edward"https://www.zbmath.org/authors/?q=ai:frenkel.edward-v"Kazhdan, David"https://www.zbmath.org/authors/?q=ai:kazhdan.david-aLet \(X\) be a smooth complex projective curve and \(G\) be a complex reductive group with Langlands dual group \({}^LG\). In its most naive form, the geometric Langlands correspondence seeks to parametrize Hecke eigensheaves on the moduli stack \(\mathrm{Bun}_{G}\) by flat \({}^LG\)-connections on \(X\).
In this paper, following a question of R. Langlands, the authors formulate a conjectural function-theoretic version of the Langlands correspondence for complex curves, more akin to the classical formulation of the Langlands correspondence as a spectral problem for Hecke operators.
Assume for simplicity that \(G\) is simple and simply-connected. In that case the canonical bundle \(K\) on \(\mathrm{Bun}_{G}\) has a square root \(K^{1/2}\). Denote by \(\overline{K}^{1/2}\) the anti-holomorphic complex conjugate of \(K^{1/2}\). The authors propose to study sections of the \(C^{\infty}\)-line bundle \(\Omega^{1/2}:=K^{1/2}\otimes \overline{K}^{1/2}\) of half-densities instead of functions on \(\mathrm{Bun}_{G}\). The line bundle \(\Omega^{1/2}\) admits an action of the algebra \(\mathcal{A} = D_{G} \otimes_{\mathbb{C}} \overline{D}_{G}\) where \(D_{G}\) is the algebra of global regular differential operators acting on \(K^{1/2}\). This algebra comes with a natural anti-linear involution and we denote by \(\mathcal{A}_{\mathbb{R}}\) the \(\mathbb{R}\)-algebra of invariants of that involution.
Denote by \(\mathrm{Bun}^{\circ}_{G}\) the coarse moduli space classifying stable \(G\)-bundles whose automorphism group is \(Z(G)\). Consider the space
\[
\mathcal{H} = L^2(\mathrm{Bun}_{G})
\]
defined as the completion of the space of smooth compactly supported sections of \(\Omega^{1/2}\) on \(\mathrm{Bun}_{G}^{\circ}\).
The authors make the following conjectures. See Conjectures 1.9.--1.11. for more details.
\begin{enumerate}
\item There is an \(\mathcal{A}\)-invariant extension \(S(\mathcal{A}) \subset \mathcal{H}\) of \(V\) such that \((\mathcal{A}_{\mathbb{R}}, S(\mathcal{A}))\) is a strongly commuting family of unbounded essentially self-adjoint operators on \(\mathcal{H}\). This allows one to define the joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\), see \S 11.
\item The joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\) is discrete. By a result of Beilinson and Drinfeld it is therefore parametrized by a countable subset \(\Sigma\) of the space of \({}^L G\)-opers on \(X\) and the joint \(\mathcal{A}\)-eigen sections form a basis of \(L^2(\mathrm{Bun}_{G})\).
\item The set \(\Sigma\) is contained in the set of \({}^LG\)-opers on \(X(\mathbb{C})\) which are defined over \(\mathbb{R}\).
\end{enumerate}
Generalizing the set-up to bundles with parabolic structures, the authors prove their conjectures in the abelian case \(G=\mathrm{GL}_1\) and in the case \(G=\mathrm{SL}_2\) and \(X=\mathbb{P}^1\) with at least four marked points. In these cases they prove that the \({}^LG\)-opers \(\Sigma\) coming from the spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) are not only contained in the set of \({}^LG\)-opers defined over \(\mathbb{R}\), but that they actually coincide.
For the entire collection see [Zbl 1461.37002].
Reviewer: Konstantin Jakob (Cambridge)Gradient property of quadratic mapshttps://www.zbmath.org/1475.140222022-01-14T13:23:02.489162Z"Karzhemanov, I. V."https://www.zbmath.org/authors/?q=ai:karzhemanov.i-vSummary: Let \(f:\mathbb{P}^3\longrightarrow\mathbb{P}^3\) be a morphism given by the linear system \(\mathcal{L}\) of quadrics. Using geometry of the Jacobian surface \(\widetilde{S}\) associated with \(\mathcal{L} \), we show that if \(\widetilde{S}\) is smooth, then \(f\) is not gradient (that is \(f\neq\text{grad}F\) for any cubic polynomial \(F\)).Commensurating actions of birational groups and groups of pseudo-automorphismshttps://www.zbmath.org/1475.140232022-01-14T13:23:02.489162Z"Cantat, Serge"https://www.zbmath.org/authors/?q=ai:cantat.serge"de Cornulier, Yves"https://www.zbmath.org/authors/?q=ai:de-cornulier.yvesPseudo-automorphisms on a variety \(X\) are birational transformations acting as regular automorphisms in codimension 1. They form a group \(\mathrm{PsAut}(X)\) with \(\mathrm{Aut}(X)\subset\mathrm{PsAut}(X)\subset\mathrm{Bir}(X)\). For smooth projective surfaces, the first inclusion is an equality; for rational varieties of dimension at least 3, both inclusions are strict; for Calabi-Yau manifolds, the second inclusion is an equality, and there are examples of such varieties such that \(\mathrm{PsAut}(X)\) is infinite while \(\mathrm{Aut}(X)\) is trivial [\textit{S. Cantat} and \textit{K. Oguiso}, Am. J. Math. 137, No. 4, 1013--1044 (2015; Zbl 1386.14147)]. A group \(G\subset \mathrm{Bir}(X)\) is (pseudo-)regularizable if it is conjugate via a birational map to a group of (pseudo-)automorphisms.
In the present paper, the authors prove the following:
Theorem 1. Let \(X\) be a projective variety over an algebraically closed field. Let \(G\) be a subgroup of \(\mathrm{Bir}(X)\). If \(G\) has Property (FW), then \(G\) is pseudo-regularizable.
Property (FW) is a fixed point property for groups acting on CAT(0) cubical complexes. Equivalently, a group satisfies Property (FW) if all commensurated subsets are transfixed for any set on which the group acts. This property is satisfied for example by discrete countable groups with Kazhdan Property (T). As the title of the article suggests, the authors look at Property (FW) from the point of view of commensurated subsets.
Furthermore, they apply Theorem 1 to classify infinite groups of birational transformations of (smooth, projective, irreducible) surfaces satisfying Property (FW): Any such group is birationally conjugate to a subgroup of the automorphism group of the projective plane, of a Hirzebruch surface \(\mathbb{F}_m\) with \(m\geq1\), or of a product of a curve with \(\mathbb{P}^1\). Moreover, in positive characteristic only the case of the projective plane can occur. This is Theorem 2.
The paper ends with four open questions. Question 10.1 asks whether every birational action of \(G\) is regularizable if \(G\) satisfies Property (FW), and it was affirmatively settled by the second named author [\textit{Y. Cornulier}, Confluentes Math. 12, No. 2, 3--10 (2021; Zbl 1460.14034)], and also by [\textit{C. Urech} and \textit{A. Lonjou}, ``Actions of Cremona groups on CAT(O) cube complexes'', Duke Math. J. (to appear)]. The other three questions still seem to be open.
Reviewer: Julia Schneider (Toulouse)The embeddings of the Heisenberg group into the Cremona grouphttps://www.zbmath.org/1475.140242022-01-14T13:23:02.489162Z"Déserti, Julie"https://www.zbmath.org/authors/?q=ai:deserti.julieSummary: In this article, we describe the embeddings of the Heisenberg group into the Cremona group.Rationality problem for norm one torihttps://www.zbmath.org/1475.140252022-01-14T13:23:02.489162Z"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinari"Yamasaki, Aiichi"https://www.zbmath.org/authors/?q=ai:yamasaki.aiichiSummary: We classify stably/retract rational norm one tori in dimension \(p-1\) where \(p\) is a prime number and in dimension up to ten with some minor exceptions.Nash multiplicity sequences and Hironaka's order functionhttps://www.zbmath.org/1475.140262022-01-14T13:23:02.489162Z"Bravo, Ana"https://www.zbmath.org/authors/?q=ai:bravo.ana-maria"Pascual-Escudero, Beatriz"https://www.zbmath.org/authors/?q=ai:pascual-escudero.beatriz"Encinas, S."https://www.zbmath.org/authors/?q=ai:encinas.santiagoFor a \(d\)-dimensional algebraic variety \(X\) over a field \(k\), the arc space \(X_{\infty}\) has been introduced in an attempt to associate with \(X\) intrinsically some space, bearing information about the geometry and invariants of \(\mathrm{Sing}(X)\) [\textit{J. F. Nash jun.}, Duke Math. J. 81, No. 1, 31--38 (1995; Zbl 0880.14010)].
When \(\mathrm{char} k = 0\) a constructive resolution could be obtained by lowering the maximal multiplicity via blow ups, obtained by stratifying the maximum multiplicity locus \(\underline{Max}\ \mathrm{mult}_X\). The Hironaka's order function \(\mathrm{ord}_P^{(d)}(X)\) at a given point \(P \in X\) is the most important tool for this. The main result in this article generalizes the results in [\textit{A. Bravo} et al., Collect. Math. 68, No. 2, 175--217 (2017; Zbl 1401.14077)], proved in characteristic 0, over perfect field of arbitrary characteristic, showing that \(\mathrm{ord}_P^{(d)}(X)\) can be read using Nash multiplicity sequence along an arc centered at \(P\) [\textit{M. Lejeune-Jalabert}, Am. J. Math. 112, No. 4, 525--568 (1990; Zbl 0743.14002); \textit{M. Hickel}, Ann. Fac. Sci. Toulouse, Math. (6) 14, No. 1, 1--50 (2005; Zbl 1076.32001)]. This means that \(\mathrm{ord}_P^{(d)}(X)\) is intrinsic to \(X\), and it could be interpreted as the rate at which the graphs of the arcs at \(P\) separate from the stratum of all points with same multiplicity as \(P\).
Given \(X\) an affine variety over \(k\) perfect, a closed point \(\xi \in X\) of maximal multiplicity \(m\), and an arc \(\varphi \in X_{\infty}\) centered at \(\xi\), we have induced arc \(\Gamma_0: \mathrm{Spec} K[[t]] \rightarrow X_0 = X \times \mathbb{A}_k^1\), centered at \(\xi_0 = (\xi, 0)\). From it we get sequence of blow-ups at points \[ X_r \rightarrow \dots \rightarrow X_1 \rightarrow X_0 = X \times \mathbb{A}^1\] \[\xi_r \mapsto \dots \ \ \ \ \xi_1 \mapsto \ \ \ \ \ \ \xi_0 \] and arcs \(\Gamma_i \in X_{i, \infty}\) centered at \(\xi_i\), such that \(\Gamma_i\) is lift of \(\Gamma_0\) by the composition of the previous \(i\) blow ups, for any \(i\).
Any such sequence of blow ups directed by \(\varphi\) defines a non-increasing sequence of numbers \[m=m_0 \geq m_1 \geq \dots m_l = m_{l+1}=\dots \geq 1\] where \(m_i\) is the multiplicity of \(\xi_i\) on \(X_i\) [Hickel, loc. cit.], and this is the Nash multiplicity sequence, viewed as the multiplicity of \(X\) along \(\varphi\). Then the persistence of \(\varphi\) in \(\underline{Max}\ \mathrm{mult}_X\) is the index at which appears the first drop below \(m\).
After blowing up at regular center contained in \(\underline{Max}\ mult_X\), the maximal multiplicity does not increase. Then one can try to approach a resolution of \(X\) by finding finite sequence of blow ups at equimultiple centers, that is \[X =X_0 \leftarrow \dots \leftarrow X_m\] such that \[\max\ \mathrm{mult}_{X_0} = \dots = \max\ \mathrm{mult}_{X_{l-1}} > \max\ \mathrm{mult}_{X_l} \] at some \(l\). Any such sequence is called simplification of the multiplicity on \(X\). Continuing iteratively one gets a variety which is regular.
In general though \(\underline{Max}\ \mathrm{mult}_X\) is not regular, so to define the centers of blow ups above one has to refine the multiplicity by considering local presentation of \(\mathrm{mult}_X\) at a point. This is a local étale embedding of a neighborhood of point in \(\underline{Max}\ \mathrm{mult}_X\) in smooth \(V\), with system of weighted equations whose zeros coincide with \(\underline{Max}\ \mathrm{mult}_X\). It is stable under blow ups at regular equimultiple centers, and for any reduced equidimensional scheme there exists such a local presentation at any point [\textit{O. E. Villamayor U.}, Adv. Math. 262, 313--369 (2014; Zbl 1295.14015)].
An algorithmic resolution requires invariants permitting to define a stratification of \(\mathrm{Sing}(\mathcal{G})\), the most important invariant in constructive resolution in characteristic 0 being the Hironaka's order function in dimension \(d\). The Rees algebras are a tool to codify local presentations, and they permit also to define Hironaka's order function. For \(V\) affine, the Rees algebra over \(V\) is \(\mathcal{G} \cong \mathcal{O}_V[f_1w^{n_1},\dots f_rw^{n_r}]\) for some elements in \(\mathcal{O}_V\), \(n_1, \dots n_r\) being the weights. Then one could define its singular locus \(\mathrm{Sing}(\mathcal{G})\), and its resolution [\textit{S. Encinas} and \textit{O. Villamayor}, in: Actas del XVI coloquio Latinoamericano de álgebra. Madrid: Revista Matemática Iberoamericana. 63--85 (2007; Zbl 1222.14030)], the latter is known to exist when \(\mathrm{char}\ k =0\). Moreover, \(\underline{Max}\ \mathrm{mult}_X = \mathrm{Sing}\mathcal{G}\). Now the problem of simplification of \(\underline{Max}\ \mathrm{mult}_X\) could be translated to finding a resolution of an appropriately chosen Rees algebra over a smooth scheme.
In some cases the resolution of \(\mathcal{G}\) is equivalent to resolution of another Rees algebra, on a smooth scheme of lower dimension, supposed to be easier to obtain. For that one needs elimination algebras', but in case of \(\mathrm{char}\ k\) positive, \(\mathrm{ord}^{(d)}(X)\) does not provide enough information, and the local presentation is not stable under transformations, so new invariants are needed to refine it. One of them is the algebra of contact of ar arc with \(\underline{Max}\ \mathrm{mult}_X\) [Bravo et al., loc. cit.], here given over an arbitrary perfect field.
Reviewer: Peter Petrov (Sofia)On singularities of threefold weighted blowupshttps://www.zbmath.org/1475.140272022-01-14T13:23:02.489162Z"Chen, Yifei"https://www.zbmath.org/authors/?q=ai:chen.yifeiSummary: We answer a conjecture raised by Caucher Birkar of singularities of weighted blowups of \(\mathbb{A}^n\) for \(n \leq 3\).Effective bounds for the number of MMP-series of a smooth threefoldhttps://www.zbmath.org/1475.140282022-01-14T13:23:02.489162Z"Martinelli, Diletta"https://www.zbmath.org/authors/?q=ai:martinelli.dilettaSummary: We prove that the number of MMP-series of a smooth projective threefold of positive Kodaira dimension and of Picard number equal to three is at most two.Instanton bundles on the flag variety \(F(0,1,2)\)https://www.zbmath.org/1475.140292022-01-14T13:23:02.489162Z"Malaspina, Francesco"https://www.zbmath.org/authors/?q=ai:malaspina.francesco"Marchesi, Simone"https://www.zbmath.org/authors/?q=ai:marchesi.simone"Pons-Llopis, Juan Francisco"https://www.zbmath.org/authors/?q=ai:pons-llopis.joanAs it is very well known, the notion of \textit{mathematical instanton bundle} on \({\mathbb P}^3\) was introduced in connection to solving an essential problem of the Yang-Mills theory. The definition and the study of instanton bundles were extended to other Fano threefolds of Picard number \(1\).
In this paper, the case of a Fano threefold of higher Picard number is considered, namely the flag variety \(F(0,1,2)\) of pairs (point, line containig the point) in a fixed \({\mathbb P}^2\). This choice is based on the paper [\textit{N. J. Hitchin}, Proc. Lond. Math. Soc. (3) 43, 133--150 (1981; Zbl 0474.14024)], as explained in the introduction. In fact, the authors continue, as they mention, the study of instanton bundles on \(F:=(F(0,1,2)\) begun by \textit{S. K. Donaldson} [Prog. Math. 60, 109--119 (1985; Zbl 0578.53023)] and \textit{N. P. Buchdahl} [J. Differ. Geom. 24, 19--52 (1986; Zbl 0586.32034)].
The authors show that each instanton bundle on \(F\) is the cohomology of a monad of a certain shape and one proves that the moduli space of stable instantons of given charge \(k\) (the charge is defined via the second Chern class) contains an irreducible component of dimension \(8k-3\), which is generically smooth. The paper ends with an interesting result about the locus in \(\hbox{Hilb}^{2t+1} (F)\) of the jumping conics of a given instanton bundle on \(F\).
Reviewer: Nicolae Manolache (Bucureşti)Quasiexcellence implies strong generationhttps://www.zbmath.org/1475.140302022-01-14T13:23:02.489162Z"Aoki, Ko"https://www.zbmath.org/authors/?q=ai:aoki.koSummary: We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of \textit{A. Bondal} and \textit{M. van den Bergh} [Mosc. Math. J. 3, No. 1, 1--36 (2003; Zbl 1135.18302)]. This simultaneously extends two results of \textit{S. B. Iyengar} and \textit{R. Takahashi} [Int. Math. Res. Not. 2016, No. 2, 499--535 (2016; Zbl 1355.13015)] and \textit{A. Neeman} [Ann. Math. (2) 193, No. 3, 689--732 (2021; Zbl 07353240)] and is new even in the affine case. The main ingredient includes Gabber's weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.The derived category of the abelian category of constructible sheaveshttps://www.zbmath.org/1475.140312022-01-14T13:23:02.489162Z"Barrett, Owen"https://www.zbmath.org/authors/?q=ai:barrett.owenSummary: We show that the triangulated category of bounded constructible complexes on an algebraic variety \(X\) over an algebraically closed field is equivalent to the bounded derived category of the abelian category of constructible sheaves on \(X\), extending a theorem of Nori to the case of positive characteristic.Quillen equivalent models for the derived category of flats and the resolution propertyhttps://www.zbmath.org/1475.140322022-01-14T13:23:02.489162Z"Estrada, Sergio"https://www.zbmath.org/authors/?q=ai:estrada.sergio"Slávik, Alexander"https://www.zbmath.org/authors/?q=ai:slavik.alexanderSummary: We investigate the assumptions under which a subclass of flat quasicoherent sheaves on a quasicompact and semiseparated scheme allows us to `mock' the homotopy category of projective modules. Our methods are based on module-theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flat quasicoherent sheaves is equivalent to the derived category of very flat quasicoherent sheaves. If, in addition, the scheme satisfies the resolution property then both derived categories are equivalent to the derived category of infinite-dimensional vector bundles. The equivalences are inferred from a Quillen equivalence between the corresponding models.Derived categories of Artin-Mumford double solidshttps://www.zbmath.org/1475.140332022-01-14T13:23:02.489162Z"Hosono, Shinobu"https://www.zbmath.org/authors/?q=ai:hosono.shinobu"Takagi, Hiromichi"https://www.zbmath.org/authors/?q=ai:takagi.hiromichiThere are at least two interesting varieties associated to a web of quadrics \(W\): the Reye congruence \(X\), which is an Enriques surface, and the Artin-Mumford quartic double solid \(Y\), which is a Fano 3-fold, famous for being unirational, but not rational.
Let \(V=\mathbb C^4\) and let \(\mathbb P^9\cong \mathbb P(S^2V^*)\) be the space of quadric surfaces. A web of quadrics is a three-dimensional linear subspace \(\mathbb P^3\cong W\subset \mathbb P(S^2V^*)\) satisfying some genericity assumptions; for example, we assume that \(W\) does not contain any quadrics of rank \(1\).
The Artin-Mumford quartic double solid \(Y\) is then defined as the double cover \(Y\to W\) branched over the locus of singular quadrics, which is a quartic in \(W\cong \mathbb P^3\). Note that the variety parametrising lines in a given quadric surface \(Q\) has two connected components if \(Q\) is smooth, but only one component if \(Q\) is singular. Hence, one can regard \(Y\) as the space of pairs \((K,Q)\) consisting of a quadric \(Q\) in \(W\) and a component \(K\) of the variety of lines on \(Q\). The Artin-Mumford quartic double solid is smooth except for 10 ordinary double points. We denote the resolution given by blowing up these points by \(\widetilde Y\to Y\).
The Reye congruence \(X\subset \operatorname*{Gr}(2,V)\) is the locus of lines in \(\mathbb P(V)\) which are contained in at least two quadrics of \(W\).
The varieties \(X\) and \(Y\) are related by the correspondence
\[
Z:=\bigl\{(\ell,Q)\mid \ell\subset Q\bigr\}\subset \operatorname*{Gr}(2,V)\times W\,.
\]
Indeed, there are maps \(Z\to \operatorname*{Gr}(2,V)\), \((\ell,Q)\mapsto \ell\) and \(Z\to Y\), \((\ell,Q)\mapsto(K_\ell,Q)\) where \(K_\ell\) is the component of the variety of lines on \(Q\) containing \([\ell]\).
When mathematicians interested in derived categories recognize two interesting varieties related by a correspondence, they often try to relate the derived categories of coherent sheaves using Fourier-Mukai functors associated to the correspondence.
A first indication that there might indeed be an strong relation between \(\mathcal D^b(\widetilde Y)\) and \(\mathcal D^b(X)\) was that both contain completely orthogonal exceptional collections of length 10. On \(\widetilde Y\), this exceptional collection is given by the structure sheaves of the fibres over the double points of \(Y\), while \(X\) has such a collection consisting of certain line bundles; see [\textit{S.\ Zube}, Math.\ Notes 61, No.\ 6, 693--699 (1997; Zbl 0933.14023)].
Going much further, Ingalls and Kuznetsov [Math.\ Ann.\ 361, No. 1--2, 107--133 (2015; Zbl 1408.14069)] described full semi-orthogonal decompositions of \(\mathcal D^b(X)\) and \(\mathcal D^b(\widetilde Y)\), such that all components of the decomposition of \(\mathcal D^b(X)\) also show up in the decomposition of \(\mathcal D^b(\widetilde Y)\). This led them to conjecture that \(\mathcal D^b(X)\) embedds as a whole into \(\mathcal D^b(\widetilde Y)\), which means that there should be a fully faithful Fourier--Mukai functor
\[
\mathcal D^b(X)\hookrightarrow \mathcal D^b(\widetilde Y)\,.
\]
In the paper under review, the authors prove this conjecture under a further genericity assumption on \(W\), namely that the locus of singular quadrics in \(W\) does not contain a line.
As one would expect, the Fourier-Mukai kernel of the fully faithful embedding \(\mathcal D^b(X)\hookrightarrow\mathcal D^b(\widetilde Y)\) is related to the correspondence \(Z\), or rather its pull back \(\widetilde Z=Z\times_Y \widetilde Y\). However, the construction is much more involved than just taking the structure or ideal sheaf. Instead, it is a family of rank 2 reflective sheaves on \(\widetilde Y\) parametrized by \(X\).
Reviewer: Andreas Krug (Hannover)On Landau-Ginzburg systems, co-tropical geometry, and \(\mathcal{D}^b(X)\) of various toric Fano manifoldshttps://www.zbmath.org/1475.140342022-01-14T13:23:02.489162Z"Jerby, Yochay"https://www.zbmath.org/authors/?q=ai:jerby.yochayLet \(X\) be a Fano manifold and denote by \(f: Y\to \mathbb{C}\) its Landau-Ginzburg mirror. If \((Y,f)\) is a Lefschetz fibration, i.e.~\(f\) has isolated non-degenerate critical points, then the homological mirror symmetry conjecture states that the bounded derived category \(\mathcal{D}^b(X)\) of coherent sheaves on \(X\) is equivalent to the Fukaya-Seidel category \(\mathcal{FS}(Y,f)\) of Lefschetz thimbles of \((Y,f)\). Note that the latter depends on choices of paths in \(\mathbb{C}\), relating a reference point in \(\mathbb{C}\) to the images of critical points of \(f\). The author of the present paper relates -- in some cases -- the critical points of \(f\) directly to full exceptional collections of line bundles of \(\mathcal{D}^b(X)\).
If \((Y,f)\) is a Lefschetz fibration, then \(\mathcal{FS}(Y,f)\) is generated by an exceptional collection of vanishing Lagrangian spheres. On the \(B\)-side, the analogous question is whether \(\mathcal{D}^b(X)\) admits a full exceptional collection of objects, or even a full exceptional collection of line bundles. It was proven by Kawamata that the bounded derived category of a smooth projective toric Deligne-Mumford stack has a full exceptional collection, albeit not necessarily in line bundles. Which ones do admit a full exceptional collection in line bundles is open.
If \(X\) is toric, then there is additional structure as \(X\) admits a moment map to its Fano polytope \(\Delta\). Moreover, \(f=f_u\) is a member of the space of Laurent polynomials whose Newton polytope is \(\Delta^\circ\), the polar dual of \(\Delta\), and with parameter \(u\). It is a general theme in mirror symmetry that the critical locus \(\mathrm{Crit}(f_u)\) recognizes a lot -- if not all -- of the complex geometry of \((Y,f_u)\).
The author of the present paper connects \(\mathrm{Crit}(f_u)\) with the \(B\)-side through a natural morphism
\[
D_u : \mathrm{Crit}(f_u) \to \mathrm{Pic}(X),
\]
which the author introduced in previous work. The main question then is under what circumstances and under what conditions on \(u\) the image of \(D_u\) consists of a full exceptional collection generating \(\mathcal{D}^b(X)\). The author proves existence for certain families of simple toric Fano manifolds. In some of theses cases the author additionally proves that \(D_u\) interacts well with the monodromy associated to loops around the critical points.
The techniques of the article pass through co-tropical geometry. Writing the critical locus \(\mathrm{Crit}(f_u)\) as an intersection of hyperplanes \(W_i\), the image of the \(W_i\) under the argument map describes some co-amoebas. A suitable limit \(u\to\infty\) of this leading to co-tropical hyperplanes is used in order to find suitable exceptional maps \(D_u\).
This article is providing another piece of the puzzle of how the full homological mirror symmetry conjecture is reduced to 0-dimensional pieces in some relatively simple examples, providing further insight into mirror symmetry.
Reviewer: Michel van Garrel (Birmingham)Exceptional collections on some fake quadricshttps://www.zbmath.org/1475.140352022-01-14T13:23:02.489162Z"Lee, Kyoung-Seog"https://www.zbmath.org/authors/?q=ai:lee.kyoung-seog"Shabalin, Timofey"https://www.zbmath.org/authors/?q=ai:shabalin.timofeySummary: We construct exceptional collections of maximal length on four families of surfaces of general type with \( p_g=q=0\) which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as their orthogonal complements.Lefschetz exceptional collections in \(S_k\)-equivariant categories of \((\mathbb{P}^n)^k\)https://www.zbmath.org/1475.140362022-01-14T13:23:02.489162Z"Mironov, Mikhail"https://www.zbmath.org/authors/?q=ai:mironov.mikhailIn this work, the author proves the existence of a rectangular Lefschetz exceptional collection inside the bounded derived category of \(S_3\)-invariant sheaves over \(\mathbb{P}^n\times\mathbb{P}^n\times\mathbb{P}^n\), when \(n=3p\) or \(n=3p+1\) and \(p\geq 1\), and for \(n=2\) the author provides a minimal Lefschetz decomposition of the derived category.
To prove these results the author combines previous fundamental work on Lefschetz exceptional collection by \textit{A. D. Elagin} [Izv. Math. 73, No. 5, 893--920 (2009; Zbl 1181.14021); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 73, No. 5, 37--66 (2009)] and \textit{A. V. Fonarev} [Izv. Math. 77, No. 5, 1044--1065 (2013; Zbl 1287.14007); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 5, 203--22 (2013)], the first lays the necessary conditions for working with group invariant derived categories and the second studies Lefschetz decompositions in the context of Grassmannians using the combinatorics of the Young diagrams, with the representation theory of the Grothendieck group \(K_\mathbb{C}\) of the derived category regarded as a \((\mathrm{GL}(K_\mathbb{C}),S_3)\)-bimodule. This last approach is responsible for providing a proof that in the case \(n=3p+2\) it is impossible to find a rectangular Lefschetz decomposition and also for finding numerical restrictions to the dimension of irreducible representations of \(K_\mathbb{C}\) with respect to the Young diagrams.
The construction of the rectangular Lefschetz collections is done following Fonarev's construction, by using the box product of the canonical line bundles on \(\mathbb{P}^n\) to produce a lexicographically ordered sequence of line bundles on \(\mathbb{P}^n\times\mathbb{P}^n\times\mathbb{P}^n\) and the Young diagrams to find a subset of this collection responsible for generating a rectangular Lefschetz collection. This rectangular Lefschetz collection generates the \(S_3\)-invariant bounded derived category as a consequence of an elegant corollary in the preliminaries.
Reviewer: Victor Pretti (Campinas)Derived categories of moduli spaces of vector bundles on curves. II.https://www.zbmath.org/1475.140372022-01-14T13:23:02.489162Z"Narasimhan, M. S."https://www.zbmath.org/authors/?q=ai:narasimhan.mudumbai-sSummary: Let \(X\) be a smooth projective curve of genus \(g\) over \(\mathbb{C}\) and Mbe the moduli space of stable vector bundle of rank \(2\) and determinant isomorphic to a fixed line bundle of degree \(1\) on \(X\). Let \(E\) be the Poincaré bundle on \(X\times M\) and \(\Phi_E:D^b(X)\rightarrow D^b(M)\) Fourier-Mukai functor defined by \(E\). It was proved in our earlier paper that \(\Phi_E\) is fully faithful for every smooth projective curve of genus \(g\ge 4\). It is proved in this present paper that the result is also true for non-hyperelliptic curves of genus \(3\). Combining known results in the case of hyperelliptic curves, one obtains that \(\Phi_E\) is fully faithful for all Xof genus \(g\ge 2\).
For part I, see [\textit{M. S. Narasimhan}, J. Geom. Phys. 122, 53--58 (2017; Zbl 1390.14099)].
For the entire collection see [Zbl 1403.11002].No phantoms in the derived category of curves over arbitrary fields, and derived characterizations of Brauer-Severi varietieshttps://www.zbmath.org/1475.140382022-01-14T13:23:02.489162Z"Novaković, Saša"https://www.zbmath.org/authors/?q=ai:novakovic.sasaSummary: We show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field \(k\). Moreover, we show that a \(n\)-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length \(n + 1\), at least in characteristic zero. We conjecture that Brauer-Severi varieties \(X\) satisfy \(\mathrm{rdi}\mathrm{m}_{\mathrm{cat}}(X) = \mathrm{ind}(X) - 1\), provided period equals index, and prove this in the case of curves, surfaces and for Brauer-Severi varieties of index at most three. We believe that the results for curves are known to the experts. We nevertheless give the proofs, adding to the literature.Lifting automorphisms on abelian varieties as derived autoequivalenceshttps://www.zbmath.org/1475.140392022-01-14T13:23:02.489162Z"Srivastava, Tanya Kaushal"https://www.zbmath.org/authors/?q=ai:srivastava.tanya-kaushalSummary: We show that on an abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as a derived autoequivalence. We also compare the deformation space of these two types of deformations.A generalization of the \(b\)-function lemmahttps://www.zbmath.org/1475.140402022-01-14T13:23:02.489162Z"Raskin, Sam"https://www.zbmath.org/authors/?q=ai:raskin.samSummary: We establish some cohomological bounds in \(D\)-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the \(b\)-function lemma for non-holonomic \(D\)-modules.Vector bundles generated by sections and amplehttps://www.zbmath.org/1475.140412022-01-14T13:23:02.489162Z"Laytimi, F."https://www.zbmath.org/authors/?q=ai:laytimi.fatima"Nahm, W."https://www.zbmath.org/authors/?q=ai:nahm.wernerSummary: Let \(E\) be a vector bundle generated by sections and \(L\) be an ample line bundle over a smooth projective variety \(X\). We give here a condition for the vanishing of Dolbeault cohomology groups of the form \(H^{p,q}(X,\mathcal{S}^{\alpha}E\otimes\wedge^{\beta}E\otimes L)\). In most cases the result is stronger than what has been obtained for the condition that \(E\) is nef and \(L\) is ample. In particular a result of Zhi Jiang is improved. To derive our result we introduce a new spectral sequence, which can be used in a broader context.Brauer groups of 1-motiveshttps://www.zbmath.org/1475.140422022-01-14T13:23:02.489162Z"Bertolin, Cristiana"https://www.zbmath.org/authors/?q=ai:bertolin.cristiana"Galluzzi, Federica"https://www.zbmath.org/authors/?q=ai:galluzzi.federicaSummary: Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group \(\mathrm{H}_{\text{èt}}^2(M,\mathbb{G}_{m,M})\) of a 1-motive \(M\), whose pull-back via the unit section \(\epsilon : S \to M\) is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of \(\mathrm{H}_{\text{èt}}^2(M, \mathbb{G}_{m,M})\) come from Azumaya algebras.Brauer groups of involution surface bundleshttps://www.zbmath.org/1475.140432022-01-14T13:23:02.489162Z"Kresch, Andrew"https://www.zbmath.org/authors/?q=ai:kresch.andrew"Tschinkel, Yuri"https://www.zbmath.org/authors/?q=ai:tschinkel.yuriSummary: We present an algorithm to compute the Brauer group of involution surface bundles over rational surfaces.Slope filtrations of \(F\)-isocrystals and logarithmic decayhttps://www.zbmath.org/1475.140442022-01-14T13:23:02.489162Z"Kramer-Miller, Joe"https://www.zbmath.org/authors/?q=ai:kramer-miller.joeSummary: Let \(k\) be a perfect field of positive characteristic and let \(X\) be a smooth irreducible quasi-compact scheme over \(k\). The Drinfeld-Kedlaya theorem states that for an irreducible \(F\)-isocrystal on \(X\), the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of \(F\)-isocrystals with \(r\)-log decay. We first show that a rank one \(F\)-isocrystal with \(r\)-log decay is overconvergent if \(r < 1\). Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld-Kedlaya theorem then follows from a patching argument.On the structure of double complexeshttps://www.zbmath.org/1475.140452022-01-14T13:23:02.489162Z"Stelzig, Jonas"https://www.zbmath.org/authors/?q=ai:stelzig.jonas-robinCet article s'intéresse à la catégorie des bicomplexes bornés d'espaces vectoriels sur un corps. Il commence (\S 1) par rappeler le résultat, qui appartient au folklore, selon lequel chaque bicomplexe possède une décomposition, essentiellement unique, en somme directe de bicomplexes indécomposables, lesquels se décrivent de façon entièrement explicite (même s'il est regrettable que l'auteur s'abstienne de donner une définition formelle des bicomplexes \textit{zigzags}, le lecteur pourra la reconstituer sans trop de difficulté). L'article s'attache ensuite (\S 2) à en dégager les conséquences en termes des suites spectrales associées aux complexes doubles, puis (\S 3) en termes d'anneaux de Grothendieck.
Le but principal de ce travail (\S 4) consiste à appliquer ces résultats d'algèbre homologique abstraite au complexe double des formes différentielles d'une variété complexe compacte. Cela permet notamment de décider si certaines notions constituent ou non des invariants biméromorphes (corollaires 28 et 29).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)Reconstructing rational stable motivic homotopy theoryhttps://www.zbmath.org/1475.140462022-01-14T13:23:02.489162Z"Garkusha, Grigory"https://www.zbmath.org/authors/?q=ai:garkusha.grigorySummary: Using a recent computation of the rational minus part of $SH(k)$ by Ananyevskiy, Levine and Panin, a theorem of Cisinski and Déglise and a version of the Röndigs and Østvær theorem, rational stable motivic homotopy theory over an infinite perfect field of characteristic different from 2 is recovered in this paper from finite Milnor-Witt correspondences in the sense of Calmès and Fasel.Geometrical symmetric powers in the motivic homotopy categoryhttps://www.zbmath.org/1475.140472022-01-14T13:23:02.489162Z"Palacios, Joe"https://www.zbmath.org/authors/?q=ai:palacios.joeSummary: Symmetric powers of quasi-projective schemes can be extended via Kan extensions to geometrical symmetric powers of motivic spaces. In this work, we study geometrical symmetric powers and compare them with various symmetric powers in the unstable and stable \(\mathbb{A}^1\)-homotopy category of schemes over a field.Diophantine geometry on curves over function fieldshttps://www.zbmath.org/1475.140482022-01-14T13:23:02.489162Z"Gasbarri, Carlo"https://www.zbmath.org/authors/?q=ai:gasbarri.carloSummary: In these notes we give a reasonably self contained proof of three of the main theorems of the Diophantine geometry of curves over function fields of characteristic zero. Let \(F\) be a function field of dimension one over the field of the complex numbers \(\mathbb{C}\) i.e. a field of transcendence degree one over \(\mathbb{C}\). Let \(X_F\) be a smooth projective curve over \(F\). We prove that: \begin{itemize} \item[--] If the genus of \(X_F\) is zero then it is isomorphic, over \(F\), to the projective line \(\mathbb{P}^1\). \item [--] If the genus of \(X_F\) is one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil). \item[--] If the genus of \(X_F\) is strictly bigger than one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) is finite (former Mordell Conjecture). \end{itemize} The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [\textit{R. Hartshorne}, ``Algebraic geometry'', Graduate Texts in Mathematics, 52, 496 p. (1977; Zbl 0367.14001)] or \textit{P. Griffiths} and \textit{J. Harris} [Principles of algebraic geometry. New York, NY: John Wiley \& Sons Ltd. (1994; Zbl 0836.14001)].
For the entire collection see [Zbl 1475.14003].Étale cohomology of rank one \(\ell \)-adic local systems in positive characteristichttps://www.zbmath.org/1475.140492022-01-14T13:23:02.489162Z"Esnault, Hélène"https://www.zbmath.org/authors/?q=ai:esnault.helene"Kerz, Moritz"https://www.zbmath.org/authors/?q=ai:kerz.moritz-cThe authors prove the non-Archimedean analogs of several results known for algebraic varieties defined over a field of characteristic zero. More precisely, Let \(X\) be a smooth projective algebraic variety defined over an algebraic closed field. The main results of the paper are:
\begin{itemize}
\item[1.] the hard Lefschetz theorem holds for any \(\ell\)-adic rank one local system on \(X\);
\item[2.] the cohomology jumping loci of any arithmetic \(\ell\)-adic constructible complexes are finite unions of torsion translated formal Lie subgroups;
\item[3.] when \(X\) is an abelian variety, any arithmetic \(\ell\)-adic perverse sheaf on \(X\) satisfies generic vanishing theorems.
\end{itemize}
For analytic local systems and constructible sheaves on a complex algebraic variety, 1 is known more generally for all semi-simple local systems by \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)]. 2 is proved by \textit{N. Budur} and \textit{B. Wang} [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 1, 227--236 (2015; Zbl 1319.14027)] based on works of Simpson. 3 is proved by \textit{C. Schnell} [Publ. Math., Inst. Hautes Étud. Sci. 121, 1--55 (2015; Zbl 1386.14079)].
The main idea of the proofs is to study certain special loci, which are closed in the deformation space of rank one \(\ell\)-adic local systems and stablized by a suitable Frobenius action. As the main technical result (Theorem 3.4), the authors prove that such special loci are always fnite unions of torsion translated formal Lie subgroups. This technical theorem is proved using formal Lie groups and generalized Fourier-Mellin transformations.
Reviewer: Botong Wang (Madison)Some remarks on Ekedahl-Oort stratificationshttps://www.zbmath.org/1475.140502022-01-14T13:23:02.489162Z"Zhang, Chao"https://www.zbmath.org/authors/?q=ai:zhang.chao.5Summary: We study independence of symplectic embeddings of the theory of Ekedahl-Oort stratifications on Shimura varieties of Hodge type, by comparing two different embeddings with a third one. The main results are as follows.
1. The Ekedahl-Oort stratification is independent of the choices of symplectic embeddings.
2. Under certain reasonable assumptions, there is certain functoriality for Ekedahl-Oort stratifications with respect to morphisms of Shimura varieties.Smooth plane curves with freely acting finite groupshttps://www.zbmath.org/1475.140512022-01-14T13:23:02.489162Z"Hayashi, Taro"https://www.zbmath.org/authors/?q=ai:hayashi.taro.1|hayashi.taroSummary: The automorphism groups of smooth plane curves of degree at least 4 is considered as a finite subgroup of the projective linear group. Using this fact, in this paper, we will classify subgroups whose actions are free of automorphism groups of smooth plane curves of degree at least 4.Large automorphism groups of ordinary curves in characteristic 2https://www.zbmath.org/1475.140522022-01-14T13:23:02.489162Z"Montanucci, Maria"https://www.zbmath.org/authors/?q=ai:montanucci.maria"Speziali, Pietro"https://www.zbmath.org/authors/?q=ai:speziali.pietroAn algebraic curve \(\mathcal{X}\) is said to be ordinary if its p-rank \(\gamma(\mathcal{X})\) (or Hasse-Witt invariant) equals its genus \(g(\mathcal{X})\). \textit{S. Nakajima} [Trans. Am. Math. Soc. 303, 595--607 (1987; Zbl 0644.14010)] proved that for ordinary curves of genus at least \(2\), the bound \[ |\mathrm{Aut}(\mathcal{X})|\leq 84g(\mathcal{X})(g(\mathcal{X})-1) \] holds for the order of the automorphism groups of \(\mathcal{X}\). It is an open question whether this bound is sharp or not, at least for sufficiently large \(g\) (up to the constant). Up to now, the closest known example to it is the so-called DGZ curve, see [\textit{M. Giulietti} et al., J. Number Theory 196, 114--138 (2019; Zbl 1451.14086)].
Under the assumption that \(\mathrm{Aut}(\mathcal{X})\) satisfies certain conditions, several improvements on the Nakajima's bound have been found. For instance, in odd characteristic, if \(\mathrm{Aut}(\mathcal{X})\) is solvable, in [\textit{G. Korchmáros} and \textit{M. Montanucci}, Algebra Number Theory 13, No. 1, 1--18 (2019; Zbl 1428.14048)] it is proved that \[ |\mathrm{Aut}(\mathcal{X})|\leq 34(g(\mathcal{X})+1)^{3/2}. \]
In the paper under review, the authors extend this result to solvable automorphism groups of algebraic curves in characteristic \(2\), under the additional hypothesis that \(g(\mathcal{X})\) is even, proving that \[ |\mathrm{Aut}(\mathcal{X})|\leq 35(g(\mathcal{X})+1)^{3/2}. \] The proof is based on a deep analysis of certain quotient curves of \(\mathcal{X}\), exploiting the constraints on the structure of the \(2\)-Sylow subgroups of \(\mathrm{Aut}(\mathcal{X})\) given by the hypothesis of even genus.
Reviewer: Marco Timpanella (Perugia)Abel-Prym maps for isotypical components of Jacobianshttps://www.zbmath.org/1475.140532022-01-14T13:23:02.489162Z"Coelho, Juliana"https://www.zbmath.org/authors/?q=ai:coelho.juliana"Abreu, Kelyane"https://www.zbmath.org/authors/?q=ai:abreu.kelyaneSummary: Let \(C\) be a smooth non-rational projective curve over the complex field \(\mathbb{C}\). If \(A\) is an abelian subvariety of the Jacobian \(J(C)\), we consider the Abel-Prym map \(\varphi_A:C\rightarrow A\) defined as the composition of the Abel map of \(C\) with the norm map of \(A\). The goal of this work is to investigate the degree of the map \(\varphi_A\) in the case where \(A\) is one of the components of an isotypical decomposition of \(J(C)\). In this case we obtain a lower bound for \(\deg(\varphi_A)\) and, under some hypotheses, also an upper bound. We then apply the results obtained to compute degrees of Abel-Prym maps in a few examples. In particular, these examples show that both bounds are sharp.On torsion of superelliptic Jacobianshttps://www.zbmath.org/1475.140542022-01-14T13:23:02.489162Z"Wawrów, Wojciech"https://www.zbmath.org/authors/?q=ai:wawrow.wojciechSummary: We prove a result describing the structure of a specific subgroup of the \(m\)-torsion of the Jacobian of a general superelliptic curve \(y^m=F(x)\), generalizing the structure theorem for the \(2\)-torsion of a hyperelliptic curve. We study existence of torsion on curves of the form \(y^q=x^p-x+a\) over finite fields of characteristic \(p\). We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over \(\mathbb{Q}\).Endomorphism algebras of abelian varieties with special reference to superelliptic Jacobianshttps://www.zbmath.org/1475.140552022-01-14T13:23:02.489162Z"Zarhin, Yuri G."https://www.zbmath.org/authors/?q=ai:zarhin.yuri-gSummary: This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line.
For the entire collection see [Zbl 1403.11002].On the Schottky problem for genus five Jacobians with a vanishing theta nullhttps://www.zbmath.org/1475.140562022-01-14T13:23:02.489162Z"Agostini, Daniele"https://www.zbmath.org/authors/?q=ai:agostini.daniele"Chua, Lynn"https://www.zbmath.org/authors/?q=ai:chua.lynnFor every genus \(g\geq4\), the dimension \(3g-3\) of the moduli space \(\mathcal{J}_{g}\) of Riemann surfaces (or equivalently Jacobians, by Torelli's Theorem) is strictly smaller that the dimension \(\frac{g(g+1)}{2}\) of the moduli space \(\mathcal{A}_{g}\) of principally polarized abelian varieties. A natural question is then to find, in various subsets of \(\mathcal{A}_{g}\) having algebro-geometric meaning, equations describing their intersection of \(\mathcal{A}_{g}\). This problem is named after Schottky, who solved it for genus 4 (where the co-dimension of \(\mathcal{J}_{g}\) in \(\mathcal{A}_{g}\) is 1). The resulting equation, as well as the (full or partial) solutions to Schottky type problems, involves theta functions, as such a function is the unique gloabl section (up to scalars) of the line bundle associated with the principal polarization.
For each principally polarized abelian variety \(B\), the associated theta function defines a theta divisor on \(B\), which is singular if the point in \(\mathcal{A}_{g}\) that is associated with \(B\) lies in a particular divisor on \(\mathcal{A}_{g}\). This divisor consists of the irreducible divisor \(\theta_{\mathrm{null}}\) parameterizing principally polarized abelian varieties whose theta divisor contains a point of order 2 (thus a vanishing even theta constant) together with another irreducible divisor, and when \(g\geq4\) if contains \(\mathcal{J}_{g}\). In addition, a singular point \(P\) determines an invariant by considering the rank of the Hessian matrix of the theta function at \(P\), and one defines the closed subset \(\theta_{\mathrm{null}}^{h}\) consisting of those points in \(\theta_{\mathrm{null}}\) in which the Hessian matrix has rank \(\leq h\). It is known that \(\mathcal{J}_{g}\cap\theta_{\mathrm{null}}\) is contained in \(\theta_{\mathrm{null}}^{3}\), with equality holding when \(g=4\). The main result of this paper is that this intersection is an irreducible component of the set \(\theta_{\mathrm{null}}^{3}\).
Recall that the Baily-Borel compactification of \(\mathcal{A}_{g}\) is obtained by adding, for each \(l \leq g\), a unique cusp that is isomorphic to \(\mathcal{A}_{l}\). Moreover, any other compactification of \(\mathcal{A}_{g}\) (in particular toroidal ones) admits a unique map to the Baily-Borel compactification, and for a toroidal compactification, the inverse image of the largest cusp \(\mathcal{A}_{g-1}\) is canonical, i.e., independent of choices that are typically required for defining toroidal compactifications. It is always isomorphic to the universal family \(\mathcal{X}_{g-1}\) over \(\mathcal{A}_{g-1}\), in which the fiber over a generic point is the Kummer variety associated with the abelian variety of dimension \(g-1\) that it parameterizes (because \(-\operatorname{Id}\) is in \(\operatorname{Sp}_{2g}(\mathbb{Z})\)). The proof of the main theorem is based on working in the partial canonical compactification \(\mathcal{A}_{g}\cup\mathcal{X}_{g-1}\), and bounding the dimension of the intersection of \(\theta_{\mathrm{null}}^{3}\) with the boundary part.
The details of this analysis involve the cover \(\mathcal{A}_{g}(2)\) obtained from the congruence subgroup of level 2 in \(\operatorname{Sp}_{2g}(\mathbb{Z})\), the Gauss map from \(\mathcal{X}_{g-1}\) to the corresponding projective tangent space, and the resulting Thom-Boardman loci, all extended to the canonical toroidal boundary component. These are applied for obtaining a lower bound for the co-dimension of an irreducible component \(\mathcal{Z}_{g}\) of \(\theta_{\mathrm{null}}^{3}\) that contains \(\mathcal{J}_{g}\cap\theta_{\mathrm{null}}\), by investigating its intersection with the universal double theta divisor \(X_{g}\) in \(\mathcal{X}_{g-1}\subseteq\mathcal{A}_{g}\cup\mathcal{X}_{g-1}\). Some of the first steps can be done for any \(g\geq4\), but the later steps work only for \(g=5\) (indeed, the authors show that if \(g\geq6\) then a certain subset of \(\mathcal{A}_{g-1}\) is no longer strictly contained in \(\mathcal{A}_{g-1}\), a condition that is crucial for the proof). The authors conclude with an example, evaluated using the Julia package for theta functions.
The paper is divided into 5 sections. Section 1 is the Introduction, with the required definitions and the statement of the main theorem. Section 2 includes the notation and the definition of the basic objects. Section 3 considers the details of the divisor \(\theta_{\mathrm{null}}\) on \(\mathcal{A}_{g}\), as well as on its inverse image in \(\mathcal{A}_{g}(2)\). Section 4 presents the partial compactification, as well as and the Thom-Boardman loci of the Gauss map. Finally, Section 5 proves the main theorem, and gives the example.
Reviewer: Shaul Zemel (Jerusalem)Asymptotic period relations for Jacobian elliptic surfaceshttps://www.zbmath.org/1475.140572022-01-14T13:23:02.489162Z"Shepherd-Barron, N. I."https://www.zbmath.org/authors/?q=ai:shepherd-barron.nicholas-iSummary: We describe the image of the locus of hyperelliptic curves of genus \(g\) under the period mapping in a neighbourhood of the diagonal locus \(\mathfrak{Diag}_g\). There is just one branch for each of the alkanes \(\text{C}_g\text{H}_{2g+2}\) of elementary organic chemistry, and each branch has a simple linear description in terms of the entries of the period matrix.
This picture is replicated for simply connected Jacobian elliptic surfaces, which form the next simplest class of algebraic surfaces after K3 and abelian surfaces. In the period domain for such surfaces of geometric genus \(g\), there is a locus \(\mathscr{W}_{1^g}\) that is analogous to \(\mathfrak{Diag}_g\), and the image of the moduli space under the period map has just one branch through \(\mathscr{W}_{1^g}\) for each alkane. Each branch is smooth and has an explicit description as a vector bundle of rank \(g-1\) over a domain that contains \(\mathscr{W}_{1^g}\).The strong maximal rank conjecture and higher rank Brill-Noether theoryhttps://www.zbmath.org/1475.140582022-01-14T13:23:02.489162Z"Cotterill, Ethan"https://www.zbmath.org/authors/?q=ai:cotterill.ethan"Gonzalo, Adrián Alonso"https://www.zbmath.org/authors/?q=ai:gonzalo.adrian-alonso"Zhang, Naizhen"https://www.zbmath.org/authors/?q=ai:zhang.naizhenFor a smooth complex curve \(C\), a line bundle \(L\) on it and a vector subspace \(V\) of \(H^0(C,L)\) let \(v_m: \hbox{Sym}^m V \rightarrow H^0(C, L^\otimes M)\) be the multiplication map. The main goal of this paper is the case \(m=2\) of a variant of the ``Strong Maximal Rank Conjecture'' (SMRC) of \textit{M. Aprodu} and \textit{G. Farkas} [Clay Math. Proc. 14, 25--50 (2011; Zbl 1248.14039)]. The ``degeneracy loci in moduli spaces of linear series, in which the quadratic multiplication map \(v_2\) fails to have maximal rank'' are called ``quadratic SMRC loci of (traditional) linear series on a general curve''. The authors show that, excepting an explicit list of cases, the quadratic SMRC loci are nonempty ``whenever the dimension of the target of \(v_2\) is at most \(6\) less than that of the domain, by verifying the positivity of the corresponding SMRC classes'' ie images of SMRC loci under the Gysin map.
The above result is used to show that for a general curve of genus \(g \in \{14, 17, 18, 19, 22, 26, 31\}\) ``the moduli space of stable rank \(2\) bundles with canonical determinant and \(k\) sections is nonempty'' when \(k\) fulfills the inequality \(3g-3 -\binom{k+1}{2} \ge 0\). This gives new evidence for the famous ``conjecture of Bertram, Feinberg and independently Mukai in higher rank Brill-Noether theory''.
One should note the careful discussion of the relations to the existent literature on the subject and the clear presentation of the main technical tools which are used.
Reviewer: Nicolae Manolache (Bucureşti)Isotropic Quot schemes of orthogonal bundles over a curvehttps://www.zbmath.org/1475.140592022-01-14T13:23:02.489162Z"Cheong, Daewoong"https://www.zbmath.org/authors/?q=ai:cheong.daewoong"Choe, Insong"https://www.zbmath.org/authors/?q=ai:choe.insong"Hitching, George H."https://www.zbmath.org/authors/?q=ai:hitching.george-hIn this paper the authors study the isotropic Quot schemes \(\text{IQ}_e(V)\) parametrizing degree \(e\) isotropic subsheaves of maximal rank of an orthogonal bundle \(V\) over a curve. Their main goal is to investigate certain properties such as the non-emptyness, irreducibility and smoothness of \(\text{IQ}_e(V)\). In this regard, they obtain results analogous to those known for the classical Quot scheme [\textit{M. Popa} and \textit{M. Roth}, Invent. Math. 152, No. 3, 625--663 (2003; Zbl 1024.14015)], and the Lagrangian Quot scheme [\textit{D. Cheong}, \textit{I. Choe}, \textit{G. H. Hitching}, ``Irreducibility of Lagrangian Quot schemes over an algebraic curve'', Preprint, \url{arXiv:1804.00052}].
Let \(C\) be a complex (smooth) projective curve. By definition a vector bundle \(V\) on \(C\) is called \textit{\(L\)-valued orthogonal} if there is a non-degenerate symmetric form \(\sigma : V \otimes V \to L\) for some line bundle \(L\). Recall that a subsheaf \(E\) of \(V\) is called \textit{isotropic} if \(\sigma|_{E \otimes E}\equiv 0 \). Let \(r=2n\) or \(2n+1\) denote the rank of \(V\). Then, for each integer \(e\), the \textit{isotropic Quot scheme} \(\text{IQ}_e(V)\) is defined by
\begin{align*}
\text{IQ}_e(V) = \{ [E \to V ] : E \text{ is isotropic of rank \(n\) and degree \(e\) in }V \}
\end{align*}
as a closed subscheme of \(\text{Quot}_{n,e}(V)\). Let \(\text{IQ}^\circ_e(V)\) be the open subscheme of isotropic subbundles. The behavior of \(\text{IQ}_e(V)\) depends, as this paper shows, on the topological type of \(V\), and even more precisely on its Stiefel-Whitney class \(w(V) \in \mathbb{Z}_2\). In particular the authors prove that in certain cases, depending on \(w(V)\) and \(e\), the closure \(\overline{\text{IQ}^\circ_e(V)}\) is either empty, or has at most two irreducible components. For example, for odd rank \(r \geq 3\), they show there is an integer \(e(V)\) such that for each \(e \leq e(V)\) with \(e \equiv w(V)\) mod 2, the locus \(\overline{\text{IQ}^\circ_e(V)}\) is nonempty, irreducible and generically smooth. Moreover they provide an explicit formula of the expected dimension of \(\text{IQ}^\circ_e(V)\). For even rank \(r\) and even degree of \(L\), it may happen that \(\text{IQ}^\circ_e(V)\) is empty. In this case, the paper shows that the scheme \(\text{IQ}_e(V)\) has components consisting entirely of non-saturated isotropic subsheaves.
The key ingredient in proving their main theorems is the use of the so-called orthogonal extensions, which are analogous to the symplectic ones considered in [\textit{D. Cheong}, \textit{I. Choe}, \textit{G. H. Hitching}, ``Irreducibility of Lagrangian Quot schemes over an algebraic curve'', Preprint, \url{arXiv:1804.00052}]. In fact, in many respects the approach is similar to the symplectic case.
Reviewer: Mihai Pavel (Vandœuvre-lès-Nancy)Corrigendum to: ``Finite generation of the algebra of type a conformal blocks via birational geometry''https://www.zbmath.org/1475.140602022-01-14T13:23:02.489162Z"Moon, Han-Bom"https://www.zbmath.org/authors/?q=ai:moon.han-bom"Yoo, Sang-Bum"https://www.zbmath.org/authors/?q=ai:yoo.sang-bumCorrects the affiliation of the second author in the authors' paper [Proc. Lond. Math. Soc. (3) 120, No. 2, 242--264 (2020; Zbl 1448.14032)].Deformations of Calabi-Yau manifolds in Fano toric varietieshttps://www.zbmath.org/1475.140612022-01-14T13:23:02.489162Z"Bini, Gilberto"https://www.zbmath.org/authors/?q=ai:bini.gilberto"Iacono, Donatella"https://www.zbmath.org/authors/?q=ai:iacono.donatellaSummary: In this article, we investigate deformations of a Calabi-Yau manifold \(Z\) in a toric variety \(F\), possibly not smooth. In particular, we prove that the forgetful morphism from the Hilbert functor \(H_Z^F\) of infinitesimal deformations of \(Z\) in \(F\) to the functor of infinitesimal deformations of \(Z\) is smooth. This implies the smoothness of \(H_Z^F\) at the corresponding point in the Hilbert scheme. Moreover, we give some examples and include some computations on the Hodge numbers of Calabi-Yau manifolds in Fano toric varieties.Algebraicity of the metric tangent cones and equivariant K-stabilityhttps://www.zbmath.org/1475.140622022-01-14T13:23:02.489162Z"Li, Chi"https://www.zbmath.org/authors/?q=ai:li.chi"Wang, Xiaowei"https://www.zbmath.org/authors/?q=ai:wang.xiaowei"Xu, Chenyang"https://www.zbmath.org/authors/?q=ai:xu.chenyangThe main achievement of this article is to completely solve Donaldson-Sun's conjecture. More precisely, for a Gromov-Hausdorff limit of a sequence of KE Fano manifolds, the metric tangent cone associated to a closed point, although defined by metric structures, only depends on algebraic structure of the singularity. The strategy is to achieve the metric tangent cone as a K-polystable degeneration of some K-semistable log Fano cone singularity via two steps. The concept of normalized volume invented by Chi Li plays the key role in this two-steps degeneration. Before this article, people have known that there exists a normalized minimizer associated to the closed point on the GH-limit with desired properties, i.e. (1) it is quasi-monomial, (2) the associated graded ring is finitely generated. Thus the first step of the two-steps degeneration is to use the finite generation to get a log Fano cone singularity (which is proved to be K-semistable). The main techniques of this article is to further prove that the K-semistable log Fans cone admits a unique K-polystable degeneration. As the metric tangent cone admits a Ricci flat Kahler cone metric, it is K-polystable. Thus they could conclude that the metric tangent cone is the unique K-polystable degeneration of the K-semistable log Fano cone induced by the normalized volume minimizer. This confirms Donaldson-Sun's conjecture.
The technical part of this article is to show the uniqueness of K-polystable degeneration. For quasi-regular log Fano cone singularity, they apply a cone construction with helps from many important progress in birational geometry (e.g running MMP, ACC of lct, etc) to show that, any two K-semistable degeneration of a K-semistable log Fano pair admit a common degeneration via two special test configurations with vanishing generalized Futaki invariants. This idea is also applied to irregular case in the article to show that any two K-semistable degeneration of a K-semistable log Fano cone admits a common degeneration via two weakly special test configurations with vanishing generalized Futaki invariants. By working out the last step of Li-Xu's special test configuration theory in log Fano cone setting, they are able to conclude that the two weakly special test configurations are special test configurations.
Reviewer: Chuyu Zhou (Beijing)Deformations of rational curves in positive characteristichttps://www.zbmath.org/1475.140632022-01-14T13:23:02.489162Z"Ito, Kazuhiro"https://www.zbmath.org/authors/?q=ai:ito.kazuhiro"Ito, Tetsushi"https://www.zbmath.org/authors/?q=ai:ito.tetsushi"Liedtke, Christian"https://www.zbmath.org/authors/?q=ai:liedtke.christianThe paper contains a sufficient criterion for a smooth and proper surface or a higher-dimensional variety over an algebraically closed field of characteristic \(p>0\) to be separably uniruled, in particular, to have a negative Kodaira dimension. In the case of surfaces, the sufficient condition is the existence of a non-rigid rational curve whose closed singular points either have \(\delta\)-invariants less than \(\frac{p-1}{2}\), or Milnor (Jacobian) numbers less than \(p\). The authors show by an example that the strict upper bound \(\frac{p-1}{2}\) for the \(\delta\)-invariants is sharp. In the higher-dimensional case, the sufficient condition for the main statement amounts to the existence of a rational curve, whose singular points satisfy the bounds as above and which gives rise to a dominating family of rational curves, as well as to an extra requirement, which roughly speaking means that the field of functions on the family of rational curves is separable over the field of functions on the variety. The latter condition, in particular, prevents the existence of fibrations between smooth varieties, whose geometric generic fiber is not reduced. Moreover, the authors show that this separability condition cannot be removed by demonstrating examples of varieties that are not separably uniruled, but are dominated by a family of rational curves with a smooth member (where the family fails to satisfy the aforementioned separabiliy condition).
Reviewer: Eugenii I. Shustin (Tel Aviv)Algebraic hyperbolicity for surfaces In toric threefoldshttps://www.zbmath.org/1475.140642022-01-14T13:23:02.489162Z"Haase, Christian"https://www.zbmath.org/authors/?q=ai:haase.christian-alexander"Ilten, Nathan"https://www.zbmath.org/authors/?q=ai:ilten.nathan-owenA projective surface \(S\) is algebraically hyperbolic if the geometric genus of irreducible curves \(C\subset S\) is bounded below by an expresssion which depends on the degree of \(C\). When the surface moves in a huge family, the structure of divisors in \(S\) is expected to become quite simple for (very) general elements of the family, so it is natural to expect algebraic hyperbolicity for very general elements of relevant families of surfaces. For instance, in the family of surfaces of fixed degree \(d\geq 4\) in \(\mathbb P^3\) very general elements \(S\) have a Picard group equal to \(\mathbb Z\), generated by a plane divisor, and a sharp analysis of singularities of curves in \(S\) proves that very general surfaces of degree \(d\geq 5\) in \(\mathbb P^3\) are algebraically hyperbolic. The authors extend the result to surfaces contained in more general toric threefolds \(X\). They use focal loci to determine a bound for the geometric genus of curves in very general elements of linear systems of surfaces in \(X\), provided that the linear systems are sufficiently positive. In particular, the authors prove that for any Gorenstein toric threefold \(X\) there exists an ample divisor \(H_0\) such that for any numerically effective divisor \(D\) the very general surface in the system \(|H_0+D|\) is algebraically hyperbolic. The authors also use focal loci to study in more details the cases where \(X= \mathbb P^2\times \mathbb P^1\), \(X=\mathbb P^1\times \mathbb P^1\times \mathbb P^1\), and \(X=\) the blow up of \(\mathbb P^3\) at one point. In all the three cases the authors provide an almost complete list of linear systems whose very general divisor is an algebraically hyperbolic surface.
Reviewer: Luca Chiantini (Siena)Cubic surfaces of characteristic twohttps://www.zbmath.org/1475.140652022-01-14T13:23:02.489162Z"Kadyrsizova, Zhibek"https://www.zbmath.org/authors/?q=ai:kadyrsizova.zhibek"Kenkel, Jennifer"https://www.zbmath.org/authors/?q=ai:kenkel.jennifer"Page, Janet"https://www.zbmath.org/authors/?q=ai:page.janet"Singh, Jyoti"https://www.zbmath.org/authors/?q=ai:singh.jyoti-prakash"Smith, Karen E."https://www.zbmath.org/authors/?q=ai:smith.karen-e"Vraciu, Adela"https://www.zbmath.org/authors/?q=ai:vraciu.adela-n"Witt, Emily E."https://www.zbmath.org/authors/?q=ai:witt.emily-eSummary: Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a ``triangle''.Unstable singular del Pezzo hypersurfaces with lower indexhttps://www.zbmath.org/1475.140662022-01-14T13:23:02.489162Z"Kim, In-Kyun"https://www.zbmath.org/authors/?q=ai:kim.in-kyun"Won, Joonyeong"https://www.zbmath.org/authors/?q=ai:won.joonyeongSummary: We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.On the multicanonical systems of quasi-elliptic surfaceshttps://www.zbmath.org/1475.140672022-01-14T13:23:02.489162Z"Katsura, Toshiyuki"https://www.zbmath.org/authors/?q=ai:katsura.toshiyuki"Saito, Natsuo"https://www.zbmath.org/authors/?q=ai:saito.natsuoSummary: We consider the multicanonical systems \(|mK_S|\) of quasi-elliptic surfaces with Kodaira dimension 1 in characteristic 2. We show that for any \(m \geq 6 |mK_S|\) gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.Area in real \(K3\)-surfaceshttps://www.zbmath.org/1475.140682022-01-14T13:23:02.489162Z"Itenberg, Ilia"https://www.zbmath.org/authors/?q=ai:itenberg.ilia-v"Mikhalkin, Grigory"https://www.zbmath.org/authors/?q=ai:mikhalkin.grigory-bLet \(X\) be a minimal \(K3\)-surface and \(\sigma\colon X\to X\) a real structure, i.e., an anti-holomorphic involution. Upon rescaling the holomorphic \(2\)-form \(\omega\) on \(X\), one can assume that \(\alpha:=\operatorname{Re}\omega\) is \(\sigma^*\)-invariant whereas \(\operatorname{Im}\omega\) is skew-invariant. Then \(\alpha\) restricts to a non-vanishing \(2\)-form on the real part \(\mathbb{R}X\), which can be regarded as an area form. In particular, \(\alpha\) defines an orientation and one can speak about the areas \(A(K):=\bigl\langle\alpha,[K]\bigr\rangle\) of connected components \(K\subset\mathbb{R}X\). These areas are defined up to a common positive factor and, thus, can be compared.
A polarization of genus \(g\ge1\) is a linear system \(|C|\) containing a smooth real curve \(C\) of genus \(g\). (It is not immediately clear whether exceptional divisors or hyperelliptic polarizations are allowed.) A polarization is non-vanishing if \([\mathbb{R}C]\ne0\) in \(H_1(\mathbb{R}X;\mathbb{Z}/2)\). This implies the presence of at least one non-spherical component \(N\subset\mathbb{R}X\). Assuming that the other components \(\Sigma_1,\ldots,\Sigma_{a-1}\) are spherical and their number \(a-1\ge g\), the authors prove the following inequality: \[ A(N)>\sum_{j=1}^{a-g}A(\Sigma_j). \] An important technical tool developed in the paper and used for the degenerate case \(g=1\) is an extension to \(K3\)-surfaces of the theory of simple Harnack curves (originally introduced for the plane and, more generally, toric varieties).
Reviewer: Alex Degtyarev (Ankara)Enriques involutions on singular \(K3\) surfaces of small discriminantshttps://www.zbmath.org/1475.140692022-01-14T13:23:02.489162Z"Shimada, Ichiro"https://www.zbmath.org/authors/?q=ai:shimada.ichiro"Veniani, Davide Cesare"https://www.zbmath.org/authors/?q=ai:veniani.davide-cesareLet \(X\) be a singular \(K3\) surface with the Picard lattice \(S_X\), \(\omega_X\) a generator of \(H^{2,0}(X)\), and \(N_X\) the nef chamber. The aim of the paper is to classify Enriques involutions on X as elements in the group \(\mathrm{aut}(X):=O(S_X,\, N_X)\cap O(S_X,\,\omega_X)\).
The first step is an interpretation of the set of all Enriques involutions on \(X\) modulo conjugate in \(\mathrm{aut}(X)\) in terms of the disjoint union of double cosets of \(O(q(S_X))\).
The second step is to apply Borcherds' method in order to calculate the group \(\mathrm{aut}(X)\) and the action of \(\mathrm{aut}(X)\) on the nef chamber \(N_X\). The method can be applied once the \(K3\) surfaces are of so-called simple Borcherds type.
An algorithm of the computation is given in section 5 as a third and last step applied for \(K3\) surfaces of simple Borchers type admitting a primitive embedding of the Picard lattice into the even unimodular hyperbolic lattice of rank \(26\) and a requirement that the representation of the automorphism group of \(X\) into \(O(S_X,\,\mathcal{P}_X)\) is injective.
It turns out to be that there are 11 cases which admit Enriques involutions on singular \(K3\) surfaces with discriminant \(\leq 36\) as the authors explicitly give in a list. They conclude their work by giving a Nikulin-Kondo type realisation for \(K3\) surfaces in the list.
Reviewer: Makiko Mase (Mannheim)Explicit equations of a fake projective planehttps://www.zbmath.org/1475.140702022-01-14T13:23:02.489162Z"Borisov, Lev A."https://www.zbmath.org/authors/?q=ai:borisov.lev-a"Keum, Jonghae"https://www.zbmath.org/authors/?q=ai:keum.jonghaeA fake projective plane is a complex algebraic surface which has the same Betti numbers as the complex projective plane but is not isomorphic to it. Such an algebraic surface is of general type and its complex conjugate is another fake projective plane. Cartwright and Steger showed that there are exactly 50 pairs of fake projective planes and described them as quotients of the unit ball in \(\mathbb{C}^2\).
The authors of this article give explicit equations of a complex pair of a fake projective plane \(\mathbb{P}^2_{\text{fake}}\) embedded in \(\mathbb{P}^9\) by its bicanonical map. The automorphism group of the considered surfaces is a non-Abelian group of order 21. The authors consider the case where the minimal resolution \(Y\) of the quotient of \(\mathbb{P}^2_{\text{fake}}\) by a Sylow 7-subgroup is an elliptic surface over \(\mathbb{P}^1\) with two multiple fibers of multiplicity 2 and 4. One key result of the paper which allows the derivation of the defining equations of \(\mathbb{P}^2_{\text{fake}}\) is the identification of the corresponding double cover \(X\) of \(Y\) with a sextic singular surface in \(\mathbb{P}^3\). The authors combine nicely several computer algebra systems to obtain the final equations and to verify several properties of \(\mathbb{P}^2_{\text{fake}}\) such as being smooth.
Reviewer: Isabel Stenger (Saarbrücken)Threefolds of Kodaira dimension onehttps://www.zbmath.org/1475.140712022-01-14T13:23:02.489162Z"Chen, Hsin-Ku"https://www.zbmath.org/authors/?q=ai:chen.hsin-kuLet \(n\) be a positive integer. \textit{C. D. Hacon} and \textit{J. McKernan} (Invent. Math. 166, No. 1, 1--25 (2006; Zbl 1121.14011)) conjectured the following: There exists a constant \(s_{n}\) such that, if \(X\) is a smooth projective variety of dimension \(n\) and Kodaira dimension \(\kappa\ge0\), then for any sufficiently divisible \(s\ge s_{n}\) the \(s\)-th pluricanonical map of \(X\) is birational to the Iitaka fibration. They proved the conjecture for \(n=\kappa\).
In the article under review, the author proves that the conjecture holds for threefolds of Kodaira dimension \(1\). In particular, the \(m\)-th pluricanonical map is birational if \(m\ge5868\) and divisible by \(12\). This result, combined with existing results, implies that the conjecture is true for any \(n\le3\).
The proof is done using a case-by-case approach. Let \(X\) be a minimal terminal threefold and let \(f:X\rightarrow C\) be the Iitaka fibration. If the genus of \(C\) is positive, the result follows by weak positivity (that is, \(\deg f_{*}\omega_{X/C}^{k}\ge0\) for any \(k>0\)), so the author assumes that the genus is zero.
In the case that the general fiber of \(f\) is a \(K3\) or an Enriques surface, he supposes that \(K_{X}\) is the pull-back of an ample \(\mathbb{Q}\)-divisor. So he finds a lower bound for the degree of this \(\mathbb{Q}\)-divisor, and this will conclude the proof. He uses the Riemann-Roch theorem for singular threefolds and an algorithm.
The case where the Iitaka fibration is an Abelian or bielliptic fibration is the most difficult. The main tool that he uses is the canonical bundle formula.
Reviewer: Giosuè Muratore (Roma)Counting points on double octic Calabi-Yau threefolds over finite fieldshttps://www.zbmath.org/1475.140722022-01-14T13:23:02.489162Z"Czarnecki, Aleksander"https://www.zbmath.org/authors/?q=ai:czarnecki.aleksanderA double octic is a double cover of \(\mathbb{P}^3\) whose branch divisor is an octic surface. It is singular exactly when the branch locus is singular. In this situation the double covering is singular as well and singularities of the double cover are in one-to-one correspondence with singularities of the branch locus. \textit{S. Cynk} and \textit{T. Szemberg} [Banach Cent. Publ. 44, 93--101 (1998; Zbl 0915.14025)] gave a sufficient condition for the existence of a crepant resolution of a double cover giving a Calabi-Yau threefold.
The aim of this paper is to give a simple method of counting points on a double octic Calabi-Yau threefold over a finite field \(\mathbb{F}_q\) based on the existence of an elliptic fibration on \(X\). As an application the modularity of some double octic Calabi-Yau threefold with \(h^{1,2}=2\) is given.
Reviewer: Dominik Burek (Kraków)Algebraic varieties with simple singularities related to some reflection groupshttps://www.zbmath.org/1475.140732022-01-14T13:23:02.489162Z"Escudero, Juan García"https://www.zbmath.org/authors/?q=ai:garcia-escudero.juanTopological mirror symmetry for rank two character varieties of surface groupshttps://www.zbmath.org/1475.140742022-01-14T13:23:02.489162Z"Mauri, Mirko"https://www.zbmath.org/authors/?q=ai:mauri.mirkoSummary: The moduli spaces of flat \(\mathrm{SL}_2\)- and \(\mathrm{PGL}_2\)-connections are known to be singular SYZ-mirror partners. We establish the equality of Hodge numbers of their intersection (stringy) cohomology. In rank two, this answers a question raised by \textit{T. Hausel} in Remark 3.30 of [Adv. Lect. Math. (ALM) 25, 29--69 (2013; Zbl 1322.14027)].Average size of the automorphism group of smooth projective hypersurfaces over finite fieldshttps://www.zbmath.org/1475.140752022-01-14T13:23:02.489162Z"Matei, Vlad"https://www.zbmath.org/authors/?q=ai:matei.vladSummary: In this paper we show that the average size of the automorphism group over \(\mathbb{F}_q\) of a smooth degree \(d\) hypersurface in \(\mathbb{P}^n_{\mathbb{F}_q}\) is equal to \(1\) as \(d \to \infty\). We also discuss some consequence of this result for the moduli space of smooth degree \(d\) hypersurfaces in \(\mathbb{P}^n\).On automorphisms and the cone conjecture for Enriques surfaces in odd characteristichttps://www.zbmath.org/1475.140762022-01-14T13:23:02.489162Z"Wang, Long"https://www.zbmath.org/authors/?q=ai:wang.long|wang.long.1Summary: We prove that, for an Enriques surface in odd characteristic, the automorphism group is finitely generated and it acts on the effective nef cone with a rational polyhedral fundamental domain. We also construct a smooth projective surface in odd characteristic which is birational to an Enriques surface and whose automorphism group is discrete but not finitely generated.Semistable sheaves with symmetric \(c_1\) on Del Pezzo surfaces of degree \(5\) and \(6\)https://www.zbmath.org/1475.140772022-01-14T13:23:02.489162Z"Abe, Takeshi"https://www.zbmath.org/authors/?q=ai:abe.takeshiThe author realizes for Del Pezzo surfaces of degree 5 and 6 the program begun by \textit{J. M. Drezet} and \textit{J. Le Potier} [Ann. Sci. Éc. Norm. Supér. (4) 18, 193--243 (1985; Zbl 0586.14007)], deepened by \textit{J. M. Drezet} [J. Reine Angew. Math. 380, 14--58 (1987; Zbl 0613.14013)] and extended by the author of the paper under review for \({\mathbb P}^1 \times {\mathbb P}^1\) [\textit{T. Abe}, Nagoya Math. J. 227, 86--159 (2017; Zbl 1415.14015)]. In this paper the following aims are realized:
\begin{itemize}
\item ``determine Chern classes of semistable sheaves with symmetric \(c_1\) on Del Pezzo surfaces of degree 5 and 6 (Theorem 4.8);
\item define the height of the moduli space of semistable sheaves with symmetric \(c_1\) (Definition 4.9);
\item show that the moduli space of height zero of semistable sheaves with symmetric \(c_1\) is isomorphic to a moduli space of representations of a certain quiver (Theorem 6.4)''.
\end{itemize}
It is interesting to note that the author explains why the case of Del Pezzo surfaces of degree 5 and 6 is a natural continuation of the cases \({\mathbb P}^2\) and \({\mathbb P}^1 \times {\mathbb P}^1\).
Reviewer: Nicolae Manolache (Bucureşti)Normal bundles on the exceptional sets of simple small resolutionshttps://www.zbmath.org/1475.140782022-01-14T13:23:02.489162Z"Du, Rong"https://www.zbmath.org/authors/?q=ai:du.rong.1"Fang, Xinyi"https://www.zbmath.org/authors/?q=ai:fang.xinyiSummary: We study the normal bundles of the exceptional sets of higher dimensional isolated simple small singularities when the Picard groups of the exceptional sets are of rank one and their normal bundles have certain good filtrations. In particular, we prove numerical inequalities satisfied by normal bundles of exceptional sets. Moreover, we generalize Laufer's results on rationality and embedding dimension of singularities to higher dimension.Non-Ulrich representation typehttps://www.zbmath.org/1475.140792022-01-14T13:23:02.489162Z"Faenzi, Daniele"https://www.zbmath.org/authors/?q=ai:faenzi.daniele"Malaspina, Francesco"https://www.zbmath.org/authors/?q=ai:malaspina.francesco"Sanna, Giangiacomo"https://www.zbmath.org/authors/?q=ai:sanna.giangiacomoInspired by similar ideas from commutative algebra, a possible approach to classify the complexity of a projective variety is in terms of their associated category of aCM sheaves. \textit{D. Eisenbud} and \textit{J. Herzog} [Math. Ann. 280, No. 2, 347--352 (1988; Zbl 0616.13011)] gave a complete classification of aCM varieties with support a finite number of aCM sheaves, the so-called \textit{CM varieties of finite representation type}. Indeed they fall into a very short list. In the next step of complexity, there exist the \textit{ CM-discrete varieties} supporting an infinite but discrete number of aCM sheaves (for instance, quadrics of corank one) and \textit{CM-tame varieties}, supporting only one-dimensional families of aCM sheaves (for instance, Atiyah's classical case of elliptic curves, see [\textit{M. F. Atiyah}, Proc. Lond. Math. Soc. (3) 7, 414--452 (1957; Zbl 0084.17305)]).
Discrete and tame aCM varieties are also classified into a very short list (see [\textit{Y. A. Drozd} and \textit{G.-M. Greuel}, J. Algebra 246, No. 1, 1--54 (2001; Zbl 1065.14041); \textit{D. Faenzi} and \textit{F. Malaspina}, Adv. Math. 310, 663--695 (2017; Zbl 1387.14117)]). It was shown in [\textit{D. Faenzi} and \textit{J. Pons-Llopis}, Épijournal de Géom. Algébr., EPIGA 5, Article 8, 37 p. (2021; Zbl 07402131)] that the rest of aCM varieties are of \textit{wild representation type}, namely they support families of arbitrarily large dimension of aCM sheaves.
Among ACM sheaves, a special role is played by \textit{Ulrich sheaves}, defined as those aCM sheaves having the maximal permitted number of global sections. In the paper under review, the authors study the contribution of Ulrich sheaves to the determination of the representation type of the underlying projective variety.
More particulary, after excluding Ulrich sheaves, they show that the two rational scrolls of degree \(4\) and the Segre product \(\mathbb{P}^1\times\mathbb{P}^2\subset \mathbb{P}^5\) become of finite CM representation type, while all other varieties keep their representation type unchanged (Theorem \(1\)). More over, they present an explicit classification of aCM sheaves on the aforementioned downgraded varieties (Theorem \(2\)).
Reviewer: Joan Pons-Llopis (Maó)On ampleness of vector bundleshttps://www.zbmath.org/1475.140802022-01-14T13:23:02.489162Z"Misra, Snehajit"https://www.zbmath.org/authors/?q=ai:misra.snehajit"Ray, Nabanita"https://www.zbmath.org/authors/?q=ai:ray.nabanitaThe paper under review gives a necessary and sufficient condition for the ampleness of a semistable holomorphic bundle with zero discriminant on a complex projective variety.
Recall that a holomorphic bundle \(E\) on a projective complex variety \(X\) is \textit{ample} if the associated tautological bundle \(\mathcal{O}_{\mathbb{P}(E)}(1)\) on the projective bundle \(\mathbb{P}(E)\) is ample, namely if \(\mathcal{O}_{\mathbb{P}(E)}(1)\) supports a smooth hermitian metric with positive curvature.
On the other hand, \(E\) is called \textit{Griffiths positive} whenever \(E\) supports on itself a smooth hermitian metric with positive curvature.
It is known that if \(E\) is Griffiths positive then \(E\) is ample and a famous conjecture states the converse. Moreover, it is also well-known that ampleness of \(E\) implies ampleness of \(det(E)\) but the converse is not true in general.
The main contribution of this paper is to close this circle of statements for semistable bundle with zero discriminant (Theorem 1): in this particular case, let \((E,h)\) be a vector bundle on a projective complex variety with hermetian metric \(h\) such that \(det(E)\) is ample. Then \((E,h)\) is Griffiths positive.
Reviewer: Joan Pons-Llopis (Maó)Coherent systems on the projective linehttps://www.zbmath.org/1475.140812022-01-14T13:23:02.489162Z"Newstead, Peter"https://www.zbmath.org/authors/?q=ai:newstead.peter-e"Teixidor i Bigas, Montserrat"https://www.zbmath.org/authors/?q=ai:teixidor-i-bigas.montserratThe authors study the existence problem for stable coherent systems on the projective line when the number of sections is larger than the rank.
Reviewer: Hossein Kheiri (Isfahan)The Fourier-Mukai transform of a universal family of stable vector bundleshttps://www.zbmath.org/1475.140822022-01-14T13:23:02.489162Z"Reede, Fabian"https://www.zbmath.org/authors/?q=ai:reede.fabianOne way of understanding an algebraic variety is by studying its bounded derived category of coherent sheaves. One way to understand a derived category is to break it down into smaller admissible subcategories using semi-orthogonal decompositions. For moduli spaces of sheaves on a variety \(X\), one approach to do this is to show that the derived category of \(X\) is an admissible subcategory of the derived category of the moduli space using the Fourier-Mukai transform induced by the universal family when it is fully faithful. This approach has been used successfully on moduli of vector bundles on curves and Hilbert schemes of points on surfaces. This paper shows the limit of this approach by showing that the Fourier-Mukai transform induced by the universal family of \(\mathcal{M}_{\mathbb{P}^2}(4,1,3)\) is not fully faithful.
Reviewer: Tim Ryan (Ann Arbor)Examples of the affinely homogeneous indefinite real hypersurfaces of the space \(\mathbb{C}^3\)https://www.zbmath.org/1475.140832022-01-14T13:23:02.489162Z"Danilov, Maksim Sergeevich"https://www.zbmath.org/authors/?q=ai:danilov.maksim-sergeevichSummary: This article is devoted to the describing of the affinely homogeneous indefinite real hypersurfaces of 3-dimensional complex space. In the work 3 types of real hypersurfaces of the space \(\mathbb{C}^3\), with indefinite Levi's form are described. For every such type the examples of families of affinely homogeneous surfaces are constructed.On the Milnor fibration for \(f(z)\bar{g}(z)\). IIhttps://www.zbmath.org/1475.140842022-01-14T13:23:02.489162Z"Oka, Mutsuo"https://www.zbmath.org/authors/?q=ai:oka.mutsuo.1|oka.mutsuo.2The author discusses the existence of Milnor fibrations for real analytic functions
\((\mathbb C^n,0) \to (\mathbb C,0)\)
of the form \(H(z,\bar z)= f(z)\bar g(z)\) for holomorphic functions \(f\) and \(g\) in
terms of their Newton polyhedra. This comprises both the \textit{spherical} Milnor
fibration
\[
\mathrm{arg} H : \partial B_\varepsilon \setminus
\left( \partial B_\varepsilon \cap H^{-1}(\{0\}) \right) \to S^1 \subset \mathbb C^*, \quad
z \mapsto \frac{H(z,\bar z)}{| H(z,\bar z)|}
\]
and the \textit{tubular} Milnor fibration
\[
H : B_\varepsilon \cap H^{-1}(D_\delta \setminus\{0\}) \to D_\delta\setminus\{0\}, \quad
z \mapsto H(z,\bar z),
\]
with \(B_\varepsilon\) the ball of radius \(\varepsilon\) around the origin in \(\mathbb C^n\)
and \(D_\delta\) the disc of radius \(\delta\) in the codomain of \(H\)
for appropriate choices of \(1\gg \varepsilon\gg \delta>0\).
Note that whenever the number of variables \(n\) is greater than two, the function \(H\) necessarily
has non-isolated singularity.
For the case that both \(f\) and \(g\) have isolated singularity at the origin,
the existence of Milnor fibrations
under some further conditions
was discussed in the author's previous article [J. Math. Soc. Japan 73, No. 2, 649--669 (2021; Zbl 07367893)].
In this paper, the functions \(f\) and \(g\) are allowed to have non-isolated singularities.
The author introduces the notion of a \textit{locally tame non-degenerate complete intersection pair}
\(\{f, g\}\) which is a technical definition in terms of the Newton diagrams of \(f\) and \(g\).
He then proves that whenever such a pair satisfies the so-called \textit{toric multiplicity condition},
the critical value \(0 \in \mathbb C\) is isolated for the restriction of \(H = f \bar g\) to some
sufficiently small ball \(B \subset \mathbb C^n\) and that, furthermore,
the function \(H\) satisfies the Thom-\(a_f\)-condition at the origin. A criterion for the
Newton diagrams of the pair \(\{f, g\}\) is given such that this toric multiplicity condition holds.
Then, the above two consequences together immediately imply the existence of the tubular
Milnor fibration. Finally, it is shown that under the same hypothesis as before,
the spherical Milnor fibration also exists and that it is topologically equivalent to the
tubular one.
Reviewer: Matthias Zach (Hannover)Grothendieck-Lefschetz and Noether-Lefschetz for bundleshttps://www.zbmath.org/1475.140852022-01-14T13:23:02.489162Z"Ravindra, G. V."https://www.zbmath.org/authors/?q=ai:ravindra.g-v"Tripathi, Amit"https://www.zbmath.org/authors/?q=ai:tripathi.amitSummary: We prove a mild strengthening of a theorem of \textit{K. Česnavičius} [Algebr. Geom. 7, No. 4, 503--511 (2020; Zbl 1450.14003)] which gives a criterion for a vector bundle on a smooth complete intersection of dimension at least \(3\) to split into a sum of line bundles. We also prove an analogous statement for bundles on a general complete intersection surface.Fibrations associated to smooth quotients of abelian varietieshttps://www.zbmath.org/1475.140862022-01-14T13:23:02.489162Z"Martinez-Nuñez, Gary"https://www.zbmath.org/authors/?q=ai:martinez-nunez.garySummary: Let \(A\) be a complex abelian variety and \(G\) a finite group of automorphisms of \(A\) fixing the origin such that \(A/G\) is smooth. The quotient \(A/G\) can be seen as a fibration over an abelian variety whose fibers are isomorphic to a product of projective spaces. We classify how the fibers are glued in the case when the fibers are isomorphic to a projective space and we prove that, in general, the quotient \(A/G\) is a fibered product of such fibrations.Explicit arithmetic on abelian varietieshttps://www.zbmath.org/1475.140872022-01-14T13:23:02.489162Z"Murty, V. Kumar"https://www.zbmath.org/authors/?q=ai:murty.vijaya-kumar"Sastry, Pramathanath"https://www.zbmath.org/authors/?q=ai:sastry.pramathanathSummary: We describe linear algebra algorithms for doing arithmetic on an abelian variety which is dual to a given abelian variety. The ideas are inspired by Khuri-Makdisi's algorithms for Jacobians of curves. Let \(\chi_0\) be the Euler characteristic of the line bundle associated with an ample divisor Hon an abelian variety A. The Hilbert scheme of effective divisors Dsuch that \(\mathscr{O}(D)\) has Hilbert polynomial \((1+t)^g\chi_0\) is a projective bundle (with fibres \(\mathbb{P}^{\chi_0-1})\) over the dual abelian variety \(\widehat{A}\) via the Abel-Jacobi map. This Hilbert scheme can be embedded in a Grassmannian, so that points on it (and hence, via the above-mentioned Abel-Jacobi map, points on \(\widehat{A})\) can be represented by matrices. Arithmetic on \(\widehat{A}\) can be worked out by using linear algebra algorithms on the representing matrices.
For the entire collection see [Zbl 1403.11002].Corrigendum to: ``The affine part of the Picard scheme''https://www.zbmath.org/1475.140882022-01-14T13:23:02.489162Z"Geisser, Thomas"https://www.zbmath.org/authors/?q=ai:geisser.thomasSummary: We give a corrected version of Theorem 3, Lemma 4, and Proposition 9 in our paper [ibid. 145, No. 2, 415--422 (2009; Zbl 1163.14025)], which are incorrect as stated (as was pointed out by O. Gabber).Detecting nilpotence and projectivity over finite unipotent supergroup schemeshttps://www.zbmath.org/1475.140892022-01-14T13:23:02.489162Z"Benson, Dave"https://www.zbmath.org/authors/?q=ai:benson.david-john"Iyengar, Srikanth B."https://www.zbmath.org/authors/?q=ai:iyengar.srikanth-b"Krause, Henning"https://www.zbmath.org/authors/?q=ai:krause.henning"Pevtsova, Julia"https://www.zbmath.org/authors/?q=ai:pevtsova.juliaLet \(k\) be a field, \(\mathrm{char}(K)=p>2\), and let \(G\) be a finite unipotent supergroup scheme over \(k\). The cohomology of \(G\) will be denoted \(H^{*,*}(G,k)\), which is isomorphic to \(\mathrm{Ext}_{kG}^{*,*}(k,k)\): the latter index in the superscript arising from the \(\mathbb Z/2\mathbb Z\)-grading. Of interest is the nilpotent elements of this cohomology group, and the authors reduce this question to one involving elementary supergroup schemes.
The main result is that \(x\in H^{*,*}(G,k)\) is nilpotent if and only if \(x_K\in H^{*,*}(G\times_k K,K)\), restricted to \(H^{*,*}(E,K)\), is nilpotent, where \(K\) is an extension of \(k\) and \(E\le G\times_k K\) is elementary. Additionally, if \(M\) is a \(kG\)-module, then \(M\) is projective if and only if the restriction of \(M\times_k K\) to \(E\) is projective. These results are then applied to finite dimensional sub-Hopf algebras of the Steenrod algebra over \(\mathbb F_p\).
Reviewer: Alan Koch (Decatur)Full level structure on some group schemeshttps://www.zbmath.org/1475.140902022-01-14T13:23:02.489162Z"Guan, Chuangtian"https://www.zbmath.org/authors/?q=ai:guan.chuangtianSummary: We give a definition of full level structure on group schemes of the form \(G\times G\), where \(G\) is a finite flat commutative group scheme of rank \(p\) over a \(\mathbb{Z}_p\)-scheme \(S\) or, more generally, a truncated \(p\)-divisible group of height 1. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.Quasi-simple finite groups of essential dimension 3https://www.zbmath.org/1475.140912022-01-14T13:23:02.489162Z"Prokhorov, Yu."https://www.zbmath.org/authors/?q=ai:prokhorov.yuri-gSummary: We classify quasi-simple finite groups of essential dimension 3.On the torus quotients of Schubert varietieshttps://www.zbmath.org/1475.140922022-01-14T13:23:02.489162Z"Bonala, Narasimha Chary"https://www.zbmath.org/authors/?q=ai:bonala.narasimha-chary"Pattanayak, Santosha Kumar"https://www.zbmath.org/authors/?q=ai:pattanayak.santosha-kumarIn this paper, the authors consider the GIT quotients of minuscule Schubert varieties for the action of a maximal torus. Let \(G\) be a semisimple simply connected complex algebraic group, \(T\) a maximal torus, \(B\supset T\) a Borel subgroup and \(P\supset B\) a parabolic subgroup. Let \(\Phi^+\) be the set of positive roots with respect to \(B\). A fundamental weight \(\omega\) is called \textit{minuscule} if \(\langle\omega,\check{\beta}\rangle\le1\) for all \(\beta\in\Phi^+\) (Definition 4.1). For \(\omega\) a minuscule weight and \(P:=P_\omega\) the associated parabolic subgroup, the flag variety \(G/P\) and the Schubert varieties in \(G/P\) are also called \textit{minuscule}. Let \({\mathcal L}_\omega\) be the homogeneous line bundle on \(G/P\) associated to \(\omega\) and \(W^P\) the associated Weyl group. For \(w\in W^P\), denote by \(X_P(w)^{ss}_T({\mathcal L}_\omega)\) the set of semistable points in the Schubert variety \(X_P(w)\) with respect to \({\mathcal L}_\omega\) for the action of \(T\). Recall also that there is a unique minimal Schubert variety \(X_P(v)\) admitting semistable points. For any \(w\in W^P\), let \(Q_w\) be the associated quiver variety (for relevant material on quivers, see section 4).
Given this setup, the authors prove that the semistable locus is contained in the smooth locus for \(X_P(w)\) if and only if \(Q_v\) contains all the essential holes of \(Q_w\) (Theorem 4.9). From now on, suppose that \(G=SL(n,{\mathbb C})\). If \(1<r<n-1\) with \(\gcd(n,r)=1\) and \(P_r\) is the maximal parabolic subgroup of \(G\) associated to the simple root \(\alpha_r\), then \(X_P(w)^{ss}_T({\mathcal L}_{\omega_r})//T\) is smooth if \(Q_v\) contains all the essential holes of \(Q_w\) (Corollary 3.5). Now let \(P\) be the parabolic subgroup of \(SL(n,{\mathbb C})\) corresonding to the highest root \(\alpha_0\). Let \({\mathcal M}\) denote the descent of \({\mathcal L}_{\alpha_0}\) to the quotient \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T\). Then (Theorem 5.1) the polarized variety \((X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})//T,{\mathcal M})\) is projectively normal and (Corollary 5.2) \(X_P(w)^{ss}_T({\mathcal L}_{\alpha_0})\) is isomophic to a projective space. Theorem 5.1 is proved using standard monomials.
Reviewer: P. E. Newstead (Liverpool)Type \(D\) quiver representation varieties, double Grassmannians, and symmetric varietieshttps://www.zbmath.org/1475.140932022-01-14T13:23:02.489162Z"Kinser, Ryan"https://www.zbmath.org/authors/?q=ai:kinser.ryan"Rajchgot, Jenna"https://www.zbmath.org/authors/?q=ai:rajchgot.jennaThe study of the representation theory of quivers from a geometric viewpoint is an useful (and well understood) approach to the theory. In this paper, the authors take a step further in their program, that seeks to exhibit the interplay between the (equivariant) geometry of the representation varieties of Dynkin quivers and the geometry of Schubert varieties in multiple flag varieties.
More precisely, let \(Q\) be a type \(D\) quiver and denote the \(\Bbbk\)-representation variety for the dimension vector \(\mathbf d\) by \(\operatorname{rep}_Q(\mathbf d)\), where \(\Bbbk\) is an algebraically closed field; Let \(G=\operatorname{GL}(\mathbf d)\) be the base change group -- \(G\) acts on \(\operatorname{rep}_Q(\mathbf d)\). The main result of this work states that there is \(\operatorname{GL}(a+b)\) such that if \(K\) denotes the group \(\operatorname{GL}(a)\times \operatorname{GL}(b)\) embedded diagonally (by blocks) on \(G\), then there exists an order-preserving injective map from the set of \(\operatorname{GL}(\mathbf d)\)-orbit closures in \(\operatorname{rep}_Q(\mathbf d)\) into the set of \(P\)-orbit closures in \(K\backslash G\) for some parabolic subgroup \(P\subset G\). Moreover, any smooth equivalence class of singularities occurring in the domain of this map also occurs in the codomain. The authors also exhibit an homomorphism between the Grothendieck groups of \(K_T(K\backslash G)\to K_{T(\mathbf d)}(\operatorname{rep}_Q(\mathbf d)\)). As a consequence of their work, using general results from the theory of spherical varieties the authors derive a result from [\textit{G. Bobiński} and \textit{G. Zwara}, Colloq. Math. 94, No. 2, 285--309 (2002; Zbl 1013.14011)]: the orbit closures in type \(D\) quiver representation are normal, Cohen-Macaulay and have rational singularities.
Reviewer: Alvaro Rittatore (Montevideo)On facets of the Newton polytope for the discriminant of the polynomial systemhttps://www.zbmath.org/1475.140942022-01-14T13:23:02.489162Z"Antipova, Irina Avgustovna"https://www.zbmath.org/authors/?q=ai:antipova.i-a"Kleshkova, Ekaterina Andreevna"https://www.zbmath.org/authors/?q=ai:kleshkova.ekaterina-andreevnaSummary: We study normal directions to facets of the Newton polytope of the discriminant of the Laurent polynomial system via the tropical approach. We use the combinatorial construction proposed by \textit{A. Dickenstein} et al. [J. Am. Math. Soc. 20, No. 4, 1111--1133 (2007; Zbl 1166.14033)] for the tropicalization of algebraic varieties admitting a parametrization by a linear map followed by a monomial map.The Newton diagram and the index for a singular point of a vector field on a plainhttps://www.zbmath.org/1475.140952022-01-14T13:23:02.489162Z"Antyushina, Irina Viktorovna"https://www.zbmath.org/authors/?q=ai:antyushina.irina-viktorovna"Bliznyakov, Nikolay Myhaylovich"https://www.zbmath.org/authors/?q=ai:bliznyakov.nikolay-myhaylovichSummary: The formula for computation of the topological index for the zero singular point of an analytical vector field on a plain in terms of Newton diagrams of field's components and Cauchy indices of real rational functions, which allows to compute the index for almost all vector fields in consequence of final quantity of arithmetical and logical operations on coefficients of vector field's components, is established in the work.Blow-ups of toric surfaces and the Mori dream space propertyhttps://www.zbmath.org/1475.140962022-01-14T13:23:02.489162Z"He, Zhuang"https://www.zbmath.org/authors/?q=ai:he.zhuangSummary: We study the question of whether the blow-ups of toric surfaces of Picard number one at the identity point of the torus are Mori dream spaces. For some of these toric surfaces, the question whether the blow-up is a Mori dream space is equivalent to countably many planar interpolation problems. We state a conjecture which generalizes a theorem of González and Karu. We give new examples of Mori dream spaces and not Mori dream spaces among these blow-ups.Toric varieties and Gröbner bases: the complete \(\mathbb{Q}\)-factorial casehttps://www.zbmath.org/1475.140972022-01-14T13:23:02.489162Z"Rossi, Michele"https://www.zbmath.org/authors/?q=ai:rossi.michele"Terracini, Lea"https://www.zbmath.org/authors/?q=ai:terracini.leaThis paper is primarily concerned with questions about complete and non-projective toric varieties. The authors note early on that many examples of complete and non-projective varieties are ad-hoc variations of the classical example of Oda, and highlight a general lack of understanding as to how non-projective varieties arise among complete toric varieties. The authors highlight the need for more examples to better understand this aspect of the theory, and given the difficulty of these constructions point to the need to adopt a computer-aided approach. To this end, two algorithms are prposed for computing complete \(\mathbb{Q}\)-factorial fans over a set of vectors.
The first algorithm proposed is inefficient but of theoretical interest, leading to a conjecture and several possible applications. The second algorithm relies on the theory of Gröbner bases, where the authors deal with toric ideals arising from complete configurations, which cannot be homogeneous. The results here lead to an efficient algorithm for computing all projective complete simplicial fans with a given 1-skeleton that is based on the computation of the Gröbner fan of the associated toric ideal.
The paper concludes by presenting several nice concrete examples and outlining many opportunities and directions for future work.
Reviewer: Jeremiah Johnson (Manchester)Toric vector bundles: GAGA and Hodge theoryhttps://www.zbmath.org/1475.140982022-01-14T13:23:02.489162Z"Stelzig, Jonas"https://www.zbmath.org/authors/?q=ai:stelzig.jonas-robinSummary: We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by \textit{D. Popovici} [``Adiabatic limit and deformations of complex structures'', \url{arXiv:1901.04087}] using analytic techniques. Both results make use of the Rees-bundle construction.Toroidal compactification: the generalised ball casehttps://www.zbmath.org/1475.140992022-01-14T13:23:02.489162Z"Kasparian, A. K."https://www.zbmath.org/authors/?q=ai:kasparian.azniv-k"Sankaran, G. K."https://www.zbmath.org/authors/?q=ai:sankaran.gregory-kumarSummary: We give an introduction to toroidal compactification in two ways: first we give an overview of the origins and uses of the construction, and then we work out many of the details in the concrete case of quotients of the generalised ball \(\mathbb{B}(a,b)=U(a,b)/(U(a)\times U(b))\).
For the entire collection see [Zbl 1475.14003].Explicit Poincaré duality in the cohomology ring of the \(\mathrm{SU}(2)\) character variety of a surfacehttps://www.zbmath.org/1475.141002022-01-14T13:23:02.489162Z"Jeffrey, Lisa"https://www.zbmath.org/authors/?q=ai:jeffrey.lisa-c"Lindberg, Aidan"https://www.zbmath.org/authors/?q=ai:lindberg.aidan"Rayan, Steven"https://www.zbmath.org/authors/?q=ai:rayan.stevenSummary: We provide an explicit description of the Poincaré duals of each generator of the rational cohomology ring of the \(\mathrm{SU}(2)\) character variety for a genus \(g\) surface with central extension -- equivalently, that of the moduli space of stable holomorphic bundles of rank 2 and odd degree.Identifiability of rank-3 tensorshttps://www.zbmath.org/1475.141012022-01-14T13:23:02.489162Z"Ballico, Edoardo"https://www.zbmath.org/authors/?q=ai:ballico.edoardo"Bernardi, Alessandra"https://www.zbmath.org/authors/?q=ai:bernardi.alessandra"Santarsiero, Pierpaola"https://www.zbmath.org/authors/?q=ai:santarsiero.pierpaolaA given complex tensor \(T \in \mathbb{C}^{(n_1+1)}\otimes ...\otimes \mathbb{C}^{(n_k+1)}\) is said to be \(identifiable\) if there is an unique way to write it as \(T = \sum_{i=1}^r v_{1,i}\otimes ...\otimes v_{k,i}\) where the \(v_{j,i}\in \mathbb{C}^{(n_j+1)}\) and the minimal \(r\) for such a decomposition to exist is called the \(rank\) (or \(tensor\) \(rank\)) of \(T\).
In this paper a complete description of what can happen for tensors of small rank (2 or 3) is given, in particular, it is shown that tensors of rank 2 are always identifiable except when they are \(2\times 2\) matrices, while for rank 3 tensors there are other non-identifiable cases besides \(3\times 3\) matrices, and such exceptions are completely described.
Reviewer: Alessandro Gimigliano (Bologna)Signed counts of real simple rational functionshttps://www.zbmath.org/1475.141022022-01-14T13:23:02.489162Z"El Hilany, Boulos"https://www.zbmath.org/authors/?q=ai:el-hilany.boulos"Rau, Johannes"https://www.zbmath.org/authors/?q=ai:rau.johannesTo find \textit{Hurwitz numbers} is to enumerate all ramified coverings of a genus \( h \) surface by a genus \( g \) surface with fixed ramification profiles over fixed branch points. These problems admit real analogs. For instance, the problem to count degree \( d \) ramified coverings of the sphere by the sphere with \( 2d-2 \) simple branch points, all real, was solved by \textit{B. Shapiro} and \textit{A. Vainshtein} [Mosc. Math. J. 3, No. 2, 647--659 (2003; Zbl 1039.58035)]. \textit{A. Cadoret}'s approach for counting real Hurwitz numbers (with \( h = 0 \)) uses the representation theory of the symmetric groups [Pac. J. Math. 219, No. 1, 53--81 (2005; Zbl 1098.12002)]. Another approach is based on a tropical correspondence theorem [\textit{H. Markwig} and \textit{J. Rau}, Math. Z. 281, No. 1-2, 501--522 (2015; Zbl 1401.14238)]. In the polynomial case \textit{I. Itenberg} and \textit{D. Zvonkine} [Comment. Math. Helv. 93, No. 3, 441--474 (2018; Zbl 1410.14043)] proved that, provided the polynomials are counted with appropriate signs, their number does not depend on the order of the branch points on the real line. Using the same approach the authors of the reviewed paper move further in this direction and solve the problem of signed counting real simple rational functions.
The authors are interested in the number of functions \( \varphi: \mathbb{C}P^1\rightarrow\mathbb{C}P^1 \) that, in an affine coordinate system, have the form \( \varphi(z)=\psi(z)/z \) where \( \psi\in\mathbb{R}[z] \), \(\deg\psi=d\), \( \psi(0)\neq0 \), and the leading coefficient of \( \psi \) is \( 1 \). They prove that, provided such functions have prescribed ramification profiles at given real critical values and are counted with appropriate signs, their number does not depend on the order of the critical values on the real line. They also prove (non-)vanishing theorems for these signed counts and study the logarithmic growth of the absolute values of these invariants when adding further simple critical values. It follows that the obtained asymptotics coincides with that of the corresponding Hurwitz number, i.e., the number of complex functions matching the same ramification data. The proof of the invariance statement as well as the asymptotics computations are based on a reduction of the functions enumeration to the enumeration of specific planar graphs.
Reviewer: Victor Zvonilov (Nizhny Novgorod)Linear pullback components of the space of codimension one foliationshttps://www.zbmath.org/1475.141032022-01-14T13:23:02.489162Z"Ferrer, V."https://www.zbmath.org/authors/?q=ai:ferrer.viviana"Vainsencher, I."https://www.zbmath.org/authors/?q=ai:vainsencher.israelThe geometry of the spaces of codimension one foliations in complex projective spaces is a rich field of research. In particular, the space of holomorphic foliations of codimension one and degree \(d\geq 2\) in \(\mathbb{P}^n\) \((n \geq 3)\) has an irreducible component whose general element can be written as a pullback \(F^*\mathcal{F}\), where \(\mathcal{F}\) is a general foliation of degree \(d\) in \(\mathbb{P}^2\) and \(F : \mathbb{P}^n \to \mathbb{P}^2\) is a general rational linear map. The author gives a polynomial formula for the degrees of such components. Scripts for Macaulay2 are provided.
Reviewer: Jesus Muciño Raymundo (Morelia)Two kinds of real lines on real del Pezzo surfaces of degree 1https://www.zbmath.org/1475.141042022-01-14T13:23:02.489162Z"Finashin, S."https://www.zbmath.org/authors/?q=ai:finashin.sergey"Kharlamov, V."https://www.zbmath.org/authors/?q=ai:kharlamov.viatcheslav-mAs is well known, the anti-bicanonical model \(X\to\mathbb{P}^3\) of a del Pezzo surface \(X\) of degree \(1\) represents \(X\) as the double cover of a quadratic cone \(Q\subset\mathbb{P}^3\) ramified at the vertex and a smooth curve \(C\) cut on \(Q\) by a cubic surface. The deck translation \(\tau\) of the covering map is called the Bertini involution. If \(X\) is real, i.e., equipped with a real structure \(c\colon X\to X\), then so are \(C\subset Q\) and \(\tau\); hence, \(c\circ\tau\) is another real structure. Thus, real del Pezzo surfaces of degree \(1\) appear in pairs \(X^\pm\) of Bertini duals.
Using the hyperelliptic structure \(X\to Q\) and corresponding Weierstraß equation, the authors represent the real part \(X_\mathbb{R}\) as the boundary of a certain \(3\)-manifold from which \(X_\mathbb{R}\) acquires a canonical \(\mathrm{Pin}^-\)-structure, regarded as a quadratic extension \(q\colon H_1(X_\mathbb{R};\mathbb{Z}/2)\to\mathbb{Z}/4\) of the intersection index. (Theorem 1.2.1 states that this form can be singled out by a certain set of axioms.) This quadratic form \(q\) is used to split real lines \(l\subset X\) into hyperbolic (\(q(l_\mathbb{R})=1\)) and elliptic (\(q(l_\mathbb{R})=-1\)), and Theorem 1.2.2 states that the signed line count within each Bertini pair is independent of the real structure: \[ h(X^+)-e(X^+)+h(X^-)-e(X^-)=16. \] (Since \(q\) is also \(\tau\)-invariant, the authors essentially speak about the signs and signed count of tritangents to \(C\).) An intermediate statement about an individual real surface \((X,c)\) is that \[ h(X)-e(X)=2\operatorname{rk}\ker\bigl[(1+c_*)\colon H_2(X;\mathbb{Z})\to H_2(X;\mathbb{Z})\bigr]-1. \]
Reviewer: Alex Degtyarev (Ankara)Segre indices and Welschinger weights as options for invariant count of real lineshttps://www.zbmath.org/1475.141052022-01-14T13:23:02.489162Z"Finashin, Sergey"https://www.zbmath.org/authors/?q=ai:finashin.sergey"Kharlamov, Viatcheslav"https://www.zbmath.org/authors/?q=ai:kharlamov.viatcheslav-mLet \(X\) be a generic real hypersurface of degree \(2n-1, n\geq 2\) in real projective space of dimension \(n+1.\) Denote by \(\mathcal N_{\mathbb C}\) the number of complex lines and by \(\mathcal N_{\mathbb R}\) the number of real lines on \(X\). The number \(\mathcal N_{\mathbb R}\) depends on the choice of \(X;\) for example since Schlaefli (1858) it is known that in the nonsingular cubic case \((n=2)\) \(\mathcal N_{\mathbb R}\) can take the values 3,7,15,27. The book that turned this type of enumerative problems a big research topic seems to be [\textit{B. Segre}, The non-singular cubic surfaces: a new method of investigation with special reference to questions of reality. Oxford: Oxford University Press (1942; Zbl 0061.36701)]. It was shown earlier by the authors [Int. Math. Res. Not. IMRN, 16 (2013; Zbl 1312.14132)] (and also by \textit{C. Okonek} and \textit{A. Teleman} [J. Reine Ang. Math. 688, 219--241 (2014; Zbl 1344.14034) ]) that for general \(n\) one as \(\mathcal N_{\mathbb R}\geq (2n-1)!!\)
The authors say they used there that a polynomial defining \(X\) yields a section of the symmetric power \( \text{Sym}^{2n-1}(\tau_{2,n+2}^*) \) of the tautological covariant vector bundle \(\tau_{2,n+2}^*\) over the real Grassmannian \(G_{\mathbb{R}}(2,n+2)\) and the zeros of this section are precisely the real lines \(l\subset X.\) The local Euler numbers \(I^e(l,X)\) sum to the total Euler number \(\mathcal{N}_{\mathbb{R}}^e\) which is independent of \(X\) and for which the value is \((2n-1)!!.\) The inequality \(\mathcal{N}_{\mathbb{R}} \geq \mathcal{N}_{\mathbb{R}}^e\) giving a lower bound for the number of real lines on the generic hypersurface \(X\) follows.
In the present paper the authors seek a direct geometric interpretation of the local indices \(I^e(l,X);\) their previous answer worked for the case \(n=2\) only, showing that \(I^e(l,X),\) the Welschinger weight \(W(l,X),\) and the Segre-index \(S(l,X)\) are equal: \(I^e(l,X)=W(l,X)\) and \(I^e(l,X)=S(l,X).\) In the current paper they overcome the difficulties with the dimension specific proof of the first equality and authors seem particularly delighted in having been able to define a Segre index for dimensions higher than two that would generalize Segre's division of lines into elliptic and hyperbolic ones.
In Section 2 a simplified version of the Welschinger weights not appealing to any auxiliary Pin structure is given. Tools are the a real version of the Birkhoff-Grothendieck theorem on the splitting of vector bundles into a direct sum of line bundles, see [\textit{M. Hazewinkel} and \textit{C. F. Martin}, J. Pure Appl. Algebra 25, 207--211 (1982; Zbl 0489.14005)]; and the homotopy class of a certain loop in \(SO(n)\) which is used to define \(W(l,X).\) For comparison with a Pin\(^-\) structure treatment of the problem of the twenty seven lines, see [\textit{R. Benedetti} and \textit{R. Silhol}, Topology 34, No. 3, 651--678 (1995; Zbl 0996.57519)].\dots
Section 3 presents the proof that \(W(l,X)=I^e(l,X).\) In a suitable real coordinate system \(u,v,x_1, \dots, x_n\) in \(P_{\mathbb R}^{n+1}\) a line \(l=(x_1=\cdots=x_n=0)\) and one can represent the hypersurfaces containing it by \(F=x_1 p_1(u,v)+ \cdots + x_n p_n(u,v) + Q(u,v,x_1, , x_n).\) Via the standard projective coordinates (the coefficients of \(F\)), the hypersurfaces define the projective space \(\mathcal{H}\) in which the hypersurfaces which are non-singular at each point of \(l\) define a Zariski open subset \(\mathcal X.\) In the projective space \( P(\mathbb C_{n-2}[u,v]\otimes \mathbb C^n)\) of all \(n\)-tuples \([p_1: \dots:p_n]\) of degree \(2n-2\) polynomials the Zariski open subset \(\mathcal P\) of \(n\)-tuples of \(p_i\) having no common root is considered and identifiable with the space of parametrized rational curves \(P^1 \ni [u:v]\mapsto [p_1:\cdots :p_n] \in P^{n-1}.\) One has then a natural projection \(\mathcal{H}\rightarrow P(\mathbb C_{n-2}[u,v]\otimes \mathbb C^n)\) and a restriction \( \text{Jet}_l^1: \mathcal{H} \rightarrow \mathcal{P}\) and defines a subvariety \(\Delta^{\mathcal P}\) of via the vanishing of a certain determinant and \(\Delta^{\mathcal X}\) as the preimage of this under \(\text{Jet}_l^1.\) In fact \(\Delta^{\mathcal X}\) is formed by those \(X\in \mathcal X\) for which the normal bundle \(N_{l,X}\) of \(l\) in \(X\) is not balanced; it has a natural stratification in terms of splitting types of \(N_l\) and the main open stratum is denoted \(\Delta_0^{\mathcal X}.\) The walls in \(\mathcal X_{\mathbb R}\) (=points of \(\mathcal X\) fixed by complex conjugation) are the top-dimensional connected components of \((\Delta_0^{\mathcal X})_{\mathbb R}.\) The first main result, Theorem 3.3.4, says \(I^e(l,X)=W(l,X)\) for all \(X\in \mathcal X_{\mathbb R}\setminus \Delta_{\mathbb R}^{\mathcal X}\) and is proved by showing that \(I^e\) and \(W\) satisfy the same wall crossing rules.
There follows a long Section 4 dedicated to consideration of multisecants to rational curves and presenting results on the so-called Castelnuovo count of multisecants. Heavily relying on this section, the definition of the Segre index follows in Section 5. It is purely algebraic like Segre' s original definition, see Finashin and Kharlamov [loc. cit, Section 4.2] , but much more involved. The equation \(I^e(l,X)=S(l,X)\) for all \(X\in \mathcal X_{\mathbb R}\setminus \Delta_{\mathbb R}^{\mathcal X}\) is again proved by showing that the function \(S\) as a function on a space of curves satisfies the same wall crossing rules as the functions \(I^e\) and \(W\) before.
Section 6 offers another viewpoint of the generalized Segre indices in particular illustrating the theory for the cases of quintic three-folds and septic four-folds and explaining how these arguments extend to general \(n\).
Reviewer: Alexander Kovačec (Coimbra)Generic torus orbit closures in Schubert varietieshttps://www.zbmath.org/1475.141062022-01-14T13:23:02.489162Z"Lee, Eunjeong"https://www.zbmath.org/authors/?q=ai:lee.eunjeong"Masuda, Mikiya"https://www.zbmath.org/authors/?q=ai:masuda.mikiyaLet \(G\) be the general linear group \(GL(n, \mathbb{C})\) and \(B\) the Borel subgroup of the upper triangle matrices in \(G\). Let \(T\) be the torus subgroup of the diagonal matrices in \(B\). Then \(T\) acts on \(G/B\), called flag variety, induced by the left multiplication on \(G\). \textit{J. B. Carrell} and \textit{A. Kurth} [J. Algebra 233, No. 1, 122--134 (2000; Zbl 1023.14023)] showed that the closure \(\bar{ \mathscr{O}}\) of a \(T\)-orbit \(\mathscr{O}\) of an element of \(G/B\) is normal and so \(\bar{\mathscr{O}}\) is a toric variety. Moreover, if \(\bar{\mathscr{O}}\) contains all \(T\)-fixed points in \(G/B\), then it is the permutohedral variety of complex dimension \(n-1\).
An element \(w\) of the symmetric group \(S_n\) on \( \{ 1, \dots, n \} \) defines the permutation matrix \([e_{w(1)} \cdots e_{w(n)}]\) where \(\{e_1, \dots, e_n \}\) is the standard column vectors in \(\mathbb{R}^n\). So one may think of \(w\) as an element of \(G\). Then the fixed point set of the action of \(T\) on \(G/B\) can be identified with \(S_n\). Let \( w \in S_n\). Then the subset \(\overline{(BwB)/B} \subseteq G/B\) is called a \textit{Schubert variety} and it is denoted by \(X_w\). The set \(X_w\) is invariant under the \(T\)-action. The author of this paper defines that an orbit in \(X_w\) is \textit{generic} if its closure contains all \(T\)-fixed points in \(X_w\).
Section 2 of this paper (under review) studies generic points in \(X_w\) and discusses the moment map \(\mu \colon G/B \longrightarrow \mathbb{R}^n\).
Section 3 of this paper studies the fan structure of the generic torus orbit closure, denoted by \(Y_w\), of a generic \(T\)-orbit in \(X_w\). Moreover, they give necessary and sufficient conditions when a point in \(X_w\) is geberic.
Section 4 of this paper studies right weak order on \(S_n\) and right weak order intervals.
Section 5 of this paper discusses the dual cone(s) of the maximal cone(s) in the fan of the generic torus orbit closure \(Y_w\) associated to the point \(wB\). Then they study when this dual is a simplicial complex. Moreover, they showed that smoothness of \(Y_w\) is same as the simplicity of the dual cone(s) associated to the point \(wB \in Y_B\).
Section 7 of this paper gives necessary and sufficient conditions for when \(Y_w\) is smooth at other fixed points \(u \leq w\). In particular, the authors get necessary and sufficient conditions when the Schubert variety \(X_w\) is a toric variety.
Section 8 of this paper introduces a polynomial \(A_w(t)\) generalizing the concept of Eulerian number. Then they showed that if \(Y_w\) is smooth then its Poincaré polynomial agrees with \(A_w(t^2)\). Moreover, they give necessary and sufficient conditions when the polynomial \(A_w(t)\) is palindromic. At the end, the authors compute Poincaré polynomial of \(Y_w\) in terms of the combinatorics of the image \(\mu (Y_w).\)
Reviewer: Soumen Sarkar (Chennai)Real and complex supersolvable line arrangements in the projective planehttps://www.zbmath.org/1475.141072022-01-14T13:23:02.489162Z"Hanumanthu, Krishna"https://www.zbmath.org/authors/?q=ai:hanumanthu.krishna"Harbourne, Brian"https://www.zbmath.org/authors/?q=ai:harbourne.brianIn the paper under review the authors study some combinatorial problems related to the geometry of supersolvable arrangements defined over the complex and real numbers. Let \(\mathcal{L} \subset \mathbb{P}^{2}_{\mathbb{C}}\) be an arrangement of \(s\) lines in the projective plane. A point \(p\) is called a modular point for \(\mathcal{L}\) if it is an intersection point with the additional property that whenever \(q\) is another crossing point, then the line through \(p\) and \(q\) is a configurational line of \(\mathcal{L}\). We say that \(\mathcal{L}\) is supersolvable if it has a modular point. The main aim of the paper under review is to provide a classification of complex supersolvable line arrangements with respect to the number of modular points. The first result of the paper tells us that if \(\mathcal{L}\) is a complex arrangement of \(s\) lines in the projective which is non-trivial (i.e. is not a pencil of lines), then it cannot have more than \(4\) modular points. Based on that observation, the authors provide a complete description of complex supersolvable line arrangements with \(3\) and \(4\) modular points. The ultimate goal of the paper under review was to verify a conjecture saying that if \(\mathcal{C}\) is a non-trivial complex supersolvable arrangement of \(s\) lines, then the number of double points \(t_{2}(\mathcal{L})\) is bounded from below by \(s/2\). It turned out that this conjecture is true, as it has been recently shown by \textit{T. Abe} in [``Double points of free projective line arrangements'', Int. Math. Res. Notices 145 (2020; \url{doi:10.1093/imrn/rnaa145})].
Here we sum up the above loose discussion by a concrete theorem. Given a supersolvable line arrangement \(\mathcal{L}\), if it has two or more modular points and they do no all have the same multiplicity, we say that \(\mathcal{L}\) is not homogeneous, but if all modular points of \(\mathcal{L}\) have the same multiplicity, we say that \(\mathcal{L}\) is homogeneous, and \(m\)-homogeneous if the common multiplicity is \(m\).
Theorem. Let \(\mathcal{L}\) be a line arrangement having \(\mu_{\mathcal{L}} > 0\) modular points over any field \(k\).
a) If \(\mathcal{L}\) is not homogeneous, then either \(\mathcal{L}\) is a near pencil or \(\mu_{\mathcal{L}} = 2\); if \(\mu_{\mathcal{L}} = 2\), then \(\mathcal{L}\) consists of \(a\geq 2\) lines through one modular point, \(b>a\) lines through the other modular point, and we have \(s = a+b-1\) lines and \((a-1)(b-1)\) double intersection points.
b) If \(\mathcal{L}\) has a modular point of multiplicity \(2\), then \(\mathcal{L}\) is a pencil of lines.
c) If \(\mathcal{L}\) is complex and homogeneous with \(m > 2\), then \(1 \leq \mu_{\mathcal{L}} \leq 4\). If \(\mu_{\mathcal{L}} \in \{3,4\}\), we have the following possibilities:
i) If \(\mu_{\mathcal{L}} = 4\), then \(s=6\), \(m=3\), the number of double points is \(3\), the number of triple points is \(4\).
ii) If \(\mu_{\mathcal{L}} = 3\), then \(m>3\) and, up to change of coordinates, \(\mathcal{L}\) consists of the lines defined by the linear factors of \[xyz(x^{m-2} - y^{m-2})(x^{m-2} - z^{m-2})(y^{m-2}-z^{m-2}),\] and hence \(s = 3m-3\), the number of double points is \(3m-6\), the number of triple points is \((m-2)^2\), and the number of \(m\)-fold points is equal to \(3\).
Reviewer: Piotr Pokora (Kraków)Complex hyperkähler structures defined by Donaldson-Thomas invariantshttps://www.zbmath.org/1475.141082022-01-14T13:23:02.489162Z"Bridgeland, Tom"https://www.zbmath.org/authors/?q=ai:bridgeland.tom"Strachan, Ian A. B."https://www.zbmath.org/authors/?q=ai:strachan.ian-a-bThe notion of a Joyce structure, introduced by \textit{T. Bridgeland} [``Geometry from Donaldson-Thomas invariants'', Preprint, \url{arXiv:1912.06504}], is a combination of geometric structures which are expected to exist on the space of stability conditions on a CY3 triangulated category, encoded by the Donaldson-Thomas invariants. The name originates in the work of \textit{D. Joyce} [Geom. Topol. 11, 667--725 (2007; Zbl 1141.14023)]. The original definition in loc.\ cit.\ is rather complicated and given in terms of non-geometric viewpoint. The paper under this review gives a more geometric reformulation of a slightly restricted class, called a strong Joyce structure on a complex manifold \(M\) (Definition 3.3). It consists of a period structure on \(M\), introduced in \S 3.1, and a complex hyperkähler structure on the holomorphic tangent bundle \(X=T M\) satisfying certain compatibility conditions. Associated to it is a locally-defined holomorphic function \(W\) on \(X\) satisfying a system of differential equations (see (1) in \S 1 and (13), (14) in \S 3.2), which is nothing but the Joyce function appearing in the original definition. Moreover, the paper discusses a relation between a quantum deformation of the above-mentioned differential equations and the Moyal quantization of Plebański's heavenly equation. It also includes a brief but valuable review of the original definition of a Joyce structure.
Reviewer: Shintaro Yanagida (Nagoya)Quasimap wall-crossings and mirror symmetryhttps://www.zbmath.org/1475.141092022-01-14T13:23:02.489162Z"Ciocan-Fontanine, Ionuţ"https://www.zbmath.org/authors/?q=ai:ciocan-fontanine.ionut"Kim, Bumsig"https://www.zbmath.org/authors/?q=ai:kim.bumsigTo fully appreciate the results of the present paper, let us first consider the case of the quintic threefold. Givental genus 0 enumerative mirror symmetry for the quintic states that, after an identification of the parameters by the mirror map, the \(I\)-function equals the \(J\)-function. The (genus 0) \(J\)-function is a generating function of genus 0 Gromov-Witten invariants of the quintic, and the \(I\)-function is built from periods of the mirror map. In 1993, Bershadsky-Cecotti-Ooguri-Vafa (BCOV) made a physics proposal of how to extend this to higher genus invariants. Namely (see Equation \((1.5.1)\) in the present article), they proposed that
\[
I_0(q)^{2g-2} \mathcal{F}^B_g(q)= \sum_{d\geq0} Q^d \langle \; \rangle^{\infty}_{g,0,d},
\]
where \(\langle \; \rangle^{\infty}_{g,0,d}\) is the Gromov--Witten invariant of curves of genus \(g\) and degree \(d\), \(Q=Q(d)\) is the mirror map, \(I_0(q)\) is the zeroth component of the \(I\)-function and \(\mathcal{F}^B_g(q)\) is the holomorphic limit of the genus \(g\) partition function of the B-model associated to the mirror family of the quintic. While there had been some progress in other (simpler) cases, a mathematically rigorous definition of \(\mathcal{F}^B_g(q)\) for the quintic was missing.
An important consequence of the main result of the present paper is that \(\mathcal{F}^B_g(q)\) is a generating function of quasimap invariants for the stability condition \(\epsilon=0+\), where the stability parameter \(\epsilon\) is an element of \((0,\infty)\) and there are walls at \(\{1/n \, : \, n\in \mathbb{Z}\}\). Going from \(\epsilon=0+\) and crossing the walls (finitely many for each curve class) leads to the Gromov-Witten invariants \(\langle \; \rangle^{\infty}_{g,0,d}\) associated to the chamber \(\epsilon\in(1,\infty)\). This thus yields the first rigorous definition of \(\mathcal{F}^B_g(q)\). Moreover, it is defined for any Calabi-Yau variety \(X\) that is realized as a smooth projective GIT quotient and the analogous result to the above formula holds if \(X\) is a complete intersection in projective space.
The main conceptual advance of the present paper is the conjectured wall-crossing formula (Conjecture 1.1 in the article) for virtual fundamental classes of quasimap invariants associated to GIT quotients. The main result of the present paper then consists of a proof of this virtual class wall-crossing formula in the case of complete intersections in projective spaces.
For complete intersection Fano varieties of index at least 2, the authors show that the wall-crossing is trivial, i.e.~the invariants are independent of \(\epsilon\). For Fano index 1, the wall-crossing is simple and explicit.
For complete intersection Calabi-Yau varieties \(X\), the author's work firmly establishes quasimap invariants as the \(B\)-model invariants of \(X\) and enumerative mirror symmetry as a consequence of wall-crossing between Gromov-Witten and quasimap invariants. Future directions likely will consist in mapping the physics BCOV theory to the quasimap theory.
For complete intersection projective varieties of general type, the wall-crossing is new and might lead to interesting future directions.
Reviewer: Michel van Garrel (Birmingham)BCOV's Feynman rule of quintic 3-foldshttps://www.zbmath.org/1475.141102022-01-14T13:23:02.489162Z"Li, Jun"https://www.zbmath.org/authors/?q=ai:li.jun.14|li.jun|li.jun.10|li.jun.11|li.jun.1|li.jun.6|li.jun.8|li.jun.3|li.jun.12|li.jun.2|li.jun.7|li.jun.13Summary: We briefly outline the proof given by Chang-Guo-Li [\textit{H.-L. Chang} et al., ``BCOV's Feynman rule of quintic 3-folds '', \url{arXiv:1810.00394}] of the Yamaguchi-Yau polynomiality and the Bershadsky-Cecotti-Ooguri-Vafa Feynman rule conjectures for the Gromov-Witten invariants of the quintic Calabi-Yau threefolds.
For the entire collection see [Zbl 1475.00108].Equivariant quantum differential equation, Stokes bases, and \(K\)-theory for a projective spacehttps://www.zbmath.org/1475.141112022-01-14T13:23:02.489162Z"Tarasov, Vitaly"https://www.zbmath.org/authors/?q=ai:tarasov.vitaly-o"Varchenko, Alexander"https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nThe quantum connection of a complex Fano variety \(X\) is a certain connection on the trivial bundle with fiber \(H^*(X,\mathbb{C})\) on \(\mathbb{P}^1\) defined in terms of quantum multiplication on \(H^*(X,\mathbb{C})\). It has two singular points, one of which is irregular and as such is subject to the Stokes phenomenon.
The classical quantum equation for projective space \(\mathbb{P}^{n-1}\) was studied by Dubrovin who in particular related the asymptotic of solutions at the regular singular point to a certain characteristic class of the tangent bundle, called the Gamma-class. On the other hand, by a result of Guzzeti [\textit{D. Guzzetti}, Commun. Math. Phys. 207, No. 2, 341--383 (1999; Zbl 0976.53094)], the Stokes matrix at the irregular singular point is related to the Gram matrix of \(\mathbb{P}^{n-1}\) defined in terms of a full exceptional collection for the derived category of coherent sheaves. For general Fano varieties this conjectural relation is known as Dubrovin's conjecture.
In this paper, the authors consider the action of the n-dimensional torus \(T=(\mathbb{C}^{\times})^{n}\) on \(\mathbb{P}^{n-1}\) and study an equivariant version of the quantum equation of \(\mathbb{P}^{n-1}\). A key point is the compatibility of the equivariant quantum equation with a system of qKZ-difference equations, see Theorem 3.1. This allows the authors to identify the space of solutions of the joint system of equations with the space of the equivariant K-theory algebra \(K_{T}(\mathbb{P}^{n-1},\mathbb{C})\) of \(\mathbb{P}^{n-1}\). This identification comes from earlier work of the authors in which they prove it for more general partial flag varieties [\textit{V. Tarasov} and \textit{A. Varchenko}, J. Geom. Phys. 142, 179--212 (2019; Zbl 1419.82020)].
Using this identification the authors study the asymptotic of solutions close to the singular points. They prove an equivariant Gamma theorem relating the asymptotics at the regular singular point of the equivariant quantum equation to an equivariant Gamma class, see Theorem 4.3., which is based on their earlier work [\textit{V. Tarasov} and \textit{A. Varchenko}, J. Geom. Phys. 142, 179--212 (2019; Zbl 1419.82020)] on partial flag varieties. At the irregular singular point the authors describe Stokes bases in terms of exceptional bases of \(K_{T}(\mathbb{P}^{n-1},\mathbb{C})\).
Reviewer: Konstantin Jakob (Cambridge)Convex algebraic geometry of curvature operatorshttps://www.zbmath.org/1475.141122022-01-14T13:23:02.489162Z"Bettiol, Renato G."https://www.zbmath.org/authors/?q=ai:bettiol.renato-g"Kummer, Mario"https://www.zbmath.org/authors/?q=ai:kummer.mario-denis"Mendes, Ricardo A. E."https://www.zbmath.org/authors/?q=ai:mendes.ricardo-a-e\(\mathcal{C}^{p}\)-parametrization in o-minimal structureshttps://www.zbmath.org/1475.141132022-01-14T13:23:02.489162Z"Kocel-Cynk, Beata"https://www.zbmath.org/authors/?q=ai:kocel-cynk.beata"Pawłucki, Wiesław"https://www.zbmath.org/authors/?q=ai:pawlucki.wieslaw"Valette, Anna"https://www.zbmath.org/authors/?q=ai:valette-stasica.annaIn this short paper a very intuitive and elementary proof of the uniform $C^p$ parametrization in general o-minimal structures is given. This parametrization was already treated by Yomdin, Gromow and Pila and Wilkie, but he authors give here a very different proof that seems truly elementary. Their approach is clearly related to the approach of Krakow school of late Prof. Łojasiewicz, as it uses induction in a way similar to the Krakow constructions of Whitney stratifications. The very intuitive and well known idea of the cell decomposition is used. The reviewer thinks it would be nice for a reader to start with Proposition 2.8.
In the spirit of Yomdin's work the estimation on the number of parametrizing maps is given in the semialgebraic context.
As the authors mention, all this brings to mind Hironaka's rectilinearization which it generalizes in a sense, the methods used bringing to mind the very interesting work of \textit{W. Pawłucki} [Bull. Pol. Acad. Sci., Math. 32, 555--560 (1984; Zbl 0574.32010)] dealing with Puiseux desingularization of subanalytic sets. The bibliography is very thorough. The paper is certainly worth reading as it stands out by the clarity of geometrical proofs that are becoming rare in modern mathematics.
Reviewer: Zofia Denkowska (Angers)Hölder-Łojasiewicz inequalities for volumes of tame objectshttps://www.zbmath.org/1475.141142022-01-14T13:23:02.489162Z"Loi, Ta Lê"https://www.zbmath.org/authors/?q=ai:loi.ta-leLet \(\mathcal{D}\) be an \(o\)-minimal structure over the real field \(\mathbb{R}\) and \(K\subset \mathbb{R}^{m}\) a compact subset. A family \((S_{t})_{t\in T}\), \(T\subset \mathbb{R}^{p}\), of subsets of \(K\) is \textit{definiable family of subsets of }\(K\) if \(T\) is a definiable set (in \(\mathcal{D}\)) and there exists a definiable set \(S\subset T\times K\) such that \(S_{t}=\{x\in K:(t,x)\in S\}\). The author gives some uniform estimates of the volumes (in the Hausdorff measures \(\mathcal{H}^{k})\) of the images (and pre-images) of such definiable families through definiable maps. For instance, for images he proves:
Let \(h:K\rightarrow \mathbb{R}^{n}\) be a continuous definable map. Let \( (S_{t})_{t\in T}\) be a definiable family of subsets of \(K\) with \(\dim S_{t}\leq k\) for all \(t\in T.\) Then there exists an odd, strictly increasing continuous definiable\ bijection \(\varphi \) from \(\mathbb{R}\) onto \(\mathbb{R }\) such that
\[
\mathcal{H}^{k}(h(S_{t}))\leq \varphi (\operatorname{diam}(S_{t}))\ \ \ \text{for all }t\in T.
\]
He gives refined estimations under additional assumptions on the \(o\)-minimal structure \(\mathcal{D}.\) Similar results are also for preimages of such families.
Reviewer: Tadeusz Krasiński (Łódź)Constructing separable Arnold snakes of Morse polynomialshttps://www.zbmath.org/1475.141152022-01-14T13:23:02.489162Z"Sorea, Miruna-Ştefana"https://www.zbmath.org/authors/?q=ai:sorea.miruna-stefanaIn general, any snake can be associated with Morse polynomials in one variable, and the number of snakes is equal to the number of topologically nonequivalent Morse polynomials (see [\textit{V. I. Arnol'd}, Russ. Math. Surv. 47, No. 1, 1 (1992; Zbl 0791.05001); translation from Usp. Mat. Nauk 47, No. 1, 3--45 (1992)]). The author considers snakes associated with alternating permutations, given by the relative positions of critical values of a Morse polynomial (cf. [\textit{S. K. Lando}, Lectures on generating functions. Transl. from the Russian by the author. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1032.05001)]), and calls them Arnold snakes. Then using the notion of separable permutation (see [\textit{P. Bose} et al., Inf. Process. Lett. 65, No. 5, 277--283 (1998; Zbl 1338.68304); \textit{S. Kitaev}, Patterns in permutations and words. Berlin: Springer (2011; Zbl 1257.68007)]) and some other combinatorial objects and considerations, she shows how to construct explicitly polynomials in one variable with preassigned critical values configurations for a special class of Arnold snakes associated with separable permutations.
The author emphasizes also that the paper is based on the first chapter of her PhD thesis [The shapes of level curves of real polynomials near strict local minima. Université de Lille (2018), \url{https://hal.archives-ouvertes.fr/tel-01909028v1}], defended at Paul Painlevé Laboratory in Lille.
Reviewer: Aleksandr G. Aleksandrov (Moskva)Toward the combinatorial component of the Jacobian conjecturehttps://www.zbmath.org/1475.141162022-01-14T13:23:02.489162Z"Loboda, A. V."https://www.zbmath.org/authors/?q=ai:loboda.alexander-vasilevich"Shipovskaja, A. V."https://www.zbmath.org/authors/?q=ai:shipovskaya.a-vSummary: The question is considered in the article about the combinatorial properties of the collection of the sums \(\{\alpha_k+\beta_\ell\}\), that are constituted by the elements of two sets of natural numbers. The connection is shown for this question with the 2-dimensional version of the well known Jacobian conjecture. The hypothesis is formulated (and in some partial cases it is proved) about one property of the sums under consideration.Locally nilpotent derivations of factorial domainshttps://www.zbmath.org/1475.141172022-01-14T13:23:02.489162Z"El Kahoui, M'Hammed"https://www.zbmath.org/authors/?q=ai:el-kahoui.mhammed"Ouali, Mustapha"https://www.zbmath.org/authors/?q=ai:ouali.mustaphaLet \(R\subset A\) be factorial domains containing \(\mathbb Q\). In the paper under review the authors give an equivalent condition in terms of locally nilpotent derivations for \(A\) to be \(R\)-isomorphic to \(R[v,w]/(cw-h(v))\), where \(0\not=c\in R\) is not a unit and \(h(v)\in R[v]\) is nonconstant modulo every prime factor of \(c\): There exists an irreducible locally nilpotent \(R\)-derivation \(\xi\) of \(A\) with ring of constants \(A^{\xi}\) equal to \(R\) and such that there exists \(c=\xi(s)\in{\mathfrak p}{\mathfrak l}=\xi(A)\cap A^{\xi}\) with the property that the ideal \(R[s]\cap cA\) of \(R[s]\) is generated by \(c\) and a polynomial \(h(s)\in R[s]\). The result implies the isomorphism of the differential rings \((A,\xi)\) and \((R[v,w]/(cw-h(v)),\delta)\) where \(\delta(v)=c\) and \(\delta(w)=\partial_vh(v)\). The authors show that an example from a paper by Daigle gives that the result does not hold if \(A\) in not factorial. In the particular case when \(R\) is a polynomial ring in one variable, the result yields a natural generalization of of a result in Masuda characterizing Danielewski hypersurfaces whose coordinate ring is factorial. Finally, the authors apply their result to the study of triangularizable locally nilpotent \(R\)-derivations of the polynomial ring in two variables over \(R\).
Reviewer: Vesselin Drensky (Sofia)Discriminant amoebas and lopsidednesshttps://www.zbmath.org/1475.141182022-01-14T13:23:02.489162Z"Forsgård, Jens"https://www.zbmath.org/authors/?q=ai:forsgard.jensSummary: We study amoebas of the principal \(A\)-determinant in relationship with the lopsidedness condition: the (co)amoeba of the principal \(A\)-determinant stratifies the space of lopsided (co)amoebas according to equivalence of order maps -- a refinement of topological equivalence. We extend \textit{L. Nilsson} and \textit{M. Passare}'s [J. Commut. Algebra 2, No. 4, 447--471 (2010; Zbl 1237.14062)] description of the coamoeba of the \(A\)-discriminant to the case when \(A\) defines a projective toric curve. As an application, we compute coefficients of transition matrices necessary when gluing local monodromy groups of \(A\)-hypergeometric systems.Toric cycles in the complement to a complex curve in \(({\mathbb{C}^\times})^2\)https://www.zbmath.org/1475.141192022-01-14T13:23:02.489162Z"Lushin, Alexey"https://www.zbmath.org/authors/?q=ai:lushin.alexey"Pochekutov, Dmitry"https://www.zbmath.org/authors/?q=ai:pochekutov.dmitrii-yuSummary: The amoeba of a complex curve in the 2-dimensional complex torus is its image under the projection onto the real parts of the logarithmic coordinates. A toric cycle in the complement to a curve is a fiber of this projection over a point in the complement to the amoeba of the curve. We consider amoebas of complex algebraic curves defined by so-called Harnack polynomials. We prove that toric cycles are homologically independent in the complement to a such curve.Simultaneous diagonalization of incomplete matrices and applicationshttps://www.zbmath.org/1475.150112022-01-14T13:23:02.489162Z"Coron, Jean-Sébastien"https://www.zbmath.org/authors/?q=ai:coron.jean-sebastien"Notarnicola, Luca"https://www.zbmath.org/authors/?q=ai:notarnicola.luca"Wiese, Gabor"https://www.zbmath.org/authors/?q=ai:wiese.gaborThe authors study computational problems for incomplete matrices.
Problem. Let \(n \geq 2\), \(t \geq 2\) and \(2 \leq p,q \leq n\) be integers. Let \(\{ U_a : 1 \leq a \leq t\}\) be diagonal matrices in \(\mathbb{Q}^{n \times n}\). Let \(\{ W_a : 1 \leq a \leq t \}\) be matrices in \(\mathbb{Q}^{p \times q}\) and \(W_0 \in \mathbb{Q}^{p \times q}\) such that \(W_0\) has full rank and there exist matrices \( P \in \mathbb{Q}^{p \times n}\) of rank \(p\) and \( Q \in \mathbb{Q}^{n \times q}\) of rank \(q\), such that \(W_0=PQ\) and \(W_a=PU_aQ\), \(1 \leq a \leq t\). The following cases are considered:
\[
\begin{array} {lll} (A) \ p=n \ \text{and} \ q=n,\quad & & (B) \ p=n \ \text{and} \ q<n, \\
(C) \ p<n \ \text{and} \ q=n, & & (D) \ p<n \ \text{and} \ p=q. \end{array}
\]
In each case, the problem is stated as follows:
(1) Given the matrices \(\{ W_a : 1 \leq a \leq t \}\), compute \(\{(u_{1,i}, \ldots, u_{t,i}) : 1 \leq i \leq n \}\), where for \(1 \leq a \leq t\), \(u_{a,1}, \ldots, u_{a,n} \in \mathbb{Q}\) are the diagonal entries of matrices \(\{ U_a : 1 \leq a \leq t\}\);
(2) Determine whether the solution is unique.
Taking into account that Problem (A) is straightforward for any \(t \geq 1\) and Problems (B) and (C) are equivalent by the symmetry in \(p\) and \(q\), the authors devise algorithms for (C) and (D).
This problem finds its motivation in cryptanalysis. The authors show how to significantly improve previously known algorithms for solving the approximate common divisor problem and ``Breaking CLT13'' cryptographic multilinear maps.
The approximation to Problem (C) consists of using the invertibility of \(Q\) and writing
\[
W_a=PU_aQ=PQQ^{-1}U_aQ=W_0Z_a,
\]
with \(Z_a=Q^{-1}U_aQ\), \(1 \leq a \leq t\). As \(W_0\) is not invertible, it is not possible to recover \(Z_a\) directly. However, this is interpreted as a system of linear equations to be solved for \(\{Z_a\}_a\). Although this system is underdetermined, exploiting the special feature that \(\{Z_a\}_a\) commute among each other leads to additional linear equations. This allows one to recover \(\{Z_a\}_a\) uniquely, and the simultaneous diagonalization eventually yields the diagonal entries of \(\{U_a\}_a\). The authors determine exact bounds on the parameters to ensure that the system has at least as many linear equations as variables, they obtain that \(p\) and \(t\) can be set as \(O(\sqrt{n})\).
Problem (D) is reduced to Problem (C) by augmenting \(Q\) with extra columns so that it becomes invertible.
Finally, the authors present concrete experiments to confirm the theoretical results.
For the entire collection see [Zbl 1452.11005].
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Finite dimensional Lie algebras in singularitieshttps://www.zbmath.org/1475.170162022-01-14T13:23:02.489162Z"Cisneros Molina, José Luis"https://www.zbmath.org/authors/?q=ai:cisneros-molina.jose-luis"Tosun, Meral"https://www.zbmath.org/authors/?q=ai:tosun.meralSummary: Complex simple Lie algebras with simply laced root systems are classified by Dynkin diagrams of type \(A_{n}\), \(D_{n}\), \(E_6\), \(E_7\), and \(E_8\). Also the dual graphs of the minimal resolution of Kleinian singularities are precisely the same aforementioned Dynkin diagrams. In this work, we recall the basic definitions and some results of the theory of complex Lie algebras and of Kleinian singularities, in order to present a relation between finite dimensional complex simple Lie algebras and the Kleinian singularities, given by a theorem by Brieskorn. We also present the extension of Brieskorn's theorem to the simple elliptic singularity \(\tilde{D}_5\).
For the entire collection see [Zbl 1470.58001].Equivariant \(K\)-theory of semi-infinite flag manifolds and the Pieri-Chevalley formulahttps://www.zbmath.org/1475.170242022-01-14T13:23:02.489162Z"Kato, Syu"https://www.zbmath.org/authors/?q=ai:kato.syu"Naito, Satoshi"https://www.zbmath.org/authors/?q=ai:naito.satoshi"Sagaki, Daisuke"https://www.zbmath.org/authors/?q=ai:sagaki.daisukeLet \(G\) be a connected, simply connected and simple algebraic group over \(\mathbb{C}\). Fix a Borel subgroup \(B\), its maximal torus \(H\) and the Weyl group \(W\). Let \(P^+\) denote the set of integral dominant weights. Consider the Grothendieck ring \(K_H(G/B)\) of the category of \(H\)-equivariant coherent sheaves on the flag variety \(G/B\). It has a basis over the Laurent polynomial ring \(K_H(\mathrm{pt})\) given by the the classes \([\mathcal{O}_{X_w}]\) of the structure sheaves of Schubert varieties \(X_w\) for \(w \in W\). The Pieri-Chevalley formula expresses the product \([\mathcal{L}_{\lambda}] [\mathcal{O}_{X_w}]\) in this basis; here \(\lambda \in P^+\) and \(\mathcal{L}_{\lambda} \longrightarrow G/B\) is the line bundle arising from the one-dimensional \(B\)-module of weight \(\lambda\).
\textit{P. Littelmann} and \textit{C. S. Seshadri} [Prog. Math. 210, 155--176 (2003; Zbl 1100.14526)] related the above linear combination to the standard monomial theory for finite-dimensional \(G\)-modules. In more details, for \(\lambda \in P^+\), the simple \(G\)-module \(L(\lambda)\) of highest weight \(\lambda\) has a basis \((p_{\pi})\) indexed by Littelmann-Seshadri paths \(\pi\) of shape \(\lambda\). Let \(\mu \in P^+\) and view \(L(\lambda+\mu)\) as a submodule of the tensor product \(L(\lambda) \otimes L(\mu)\). The standard monomial theory provides a monomial basis \((p_{\pi} p_{\eta})\) of \(L(\lambda+\mu)\), indexed by pairs of Littelmann-Seshadri paths \(\pi\) and \(\eta\) of shapes \(\lambda\) and \(\mu\) respectively, satisfying a certain standard property. Then the linear combination for \([\mathcal{L}_{\lambda}][\mathcal{O}_{X_w}]\) is encoded in the monomial bases of \(V(\lambda+\mu)\) for specific choices of \(\mu\).
Let \(U_q(\mathfrak{g}_{\mathrm{aff}})\) denote the quantum affine algebra associated to the affinization \(\mathfrak{g}_{\mathrm{aff}}\) of the Lie algebra of \(G\), and \(W_{\mathrm{aff}}\) the affine Weyl group, which is a semi-direct product of \(W\) with the integral coweight lattice of \(G\). For \(\lambda \in P^+\) and \(x \in W_{\mathrm{aff}}\), Kashiwara introduced the level zero extremal module \(V(\lambda)\) over \(U_q(\mathfrak{g}_{\mathrm{aff}})\) and its Demazure submodule \(V_x(\lambda)\) over the nilpotent subalgebra \(U_q^-(\mathfrak{g}_{\mathrm{aff}})\), and equipped both modules with compatible crystal structure. The recent works [\textit{M. Ishii} et al., Adv. Math. 290, 967--1009 (2016; Zbl 1387.17028)] and [\textit{S. Naito} and \textit{D. Sagaki}, Math. Z. 283, No. 3--4, 937--978 (2016; Zbl 1395.17029)] described the crystal structure in terms of semi-infinite Littelmann-Seshadri paths.
In the present work the authors generalize the Pieri-Chevalley formula to an ``affine version'' of \(G/B\), the semi-infinite flag manifold \(\mathbf{Q}_G\). For that purpose, the authors establish fundamental results on semi-infinite Schubert varieties \(\mathbf{Q}_G(x)\) indexed by \(x \in W_{\mathrm{aff}}\) (actually one needs to replace the integral coweight lattice by the cone). Notably, for \(\lambda \in P^+\), the structure sheaf \(\mathcal{O}_{\mathbf{Q}_G(x)}\) twisted by the line bundle \(\mathcal{L}_{\lambda}\) has vanishing higher cohomology and its space of global sections is identified with the Demazure submodule \(V_x(-w_0 \lambda)\), where \(w_0\) denotes the longest element in the Weyl group \(W\). These results enable the authors to have a good definition of K-theory of \(\mathbf{Q}_G\), equivariant with respect to the Iwarahori subgroup of \(G(\mathbb{C}[[z]])\). To express \([\mathcal{L}_{\lambda}][\mathcal{O}_{\mathbf{Q}_G(x)}]\) as a linear combination of the classes \([\mathcal{O}_{\mathbf{Q}_G(y)}]\) for \(y \in W_{\mathrm{aff}}\), as in the classical case of Littelmann-Seshadri, the authors develop a standard monomial theory for crystal bases of Demazure modules.
Reviewer: Huafeng Zhang (Villeneuve d'Ascq)The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural moduleshttps://www.zbmath.org/1475.170332022-01-14T13:23:02.489162Z"Ecker, Jill"https://www.zbmath.org/authors/?q=ai:ecker.jill"Schlichenmaier, Martin"https://www.zbmath.org/authors/?q=ai:schlichenmaier.martinThe article under review computes (in a largely elementary fashion) the (algebraic) Lie algebra cohomology spaces \(H^3(\mathfrak{g},\mathfrak{g})\) with values in the adjoint module, for the Witt- and Virasoro Lie algebras \(\mathcal{W}\) and \(\mathcal{V}\) in characteristic zero.
Here \textit{algebraic cohomology} means that the cochain spaces in the computation of the cohomology of these infinite-dimensional Lie algebras consist of all algebraic cochains and not only of the continuous cochains with respect to some topology on \(\mathcal{W}\) and \(\mathcal{V}\). The continuous cohomology (also called Gelfand-Fuchs cohomology) is known, see
[\textit{D. B. Fuks}, Cohomology of infinite-dimensional Lie algebras. Transl. from the Russian by A. B. Sosinskiĭ. New York, NY: Consultants Bureau (1986; Zbl 0667.17005)]. Nowadays, even the algebraic cohomology of vector field Lie algebras on smooth affine algebraic varieties is computed, see
[\textit{B. Hennion} and \textit{M. Kapranov}, ``Gelfand-Fuchs cohomology in algebraic geometry and factorization algebras'', Preprint, \url{arXiv:1811.05032}],
but the case of non-trivial coefficients is open.
The outcome of the (rather tedious) computation is that \(H^3(\mathcal{W}, \mathcal{W})\) is zero, while \(H^3(\mathcal{V},\mathcal{V})\) is 1-dimensional (deduced from \(H^3(\mathcal{W},\mathcal{W})\) via the Hochschild-Serre spectral sequence).
For the entire collection see [Zbl 1460.17002].
Reviewer: Friedrich Wagemann (Nantes)The tangent complex of \(K\)-theoryhttps://www.zbmath.org/1475.190032022-01-14T13:23:02.489162Z"Hennion, Benjamin"https://www.zbmath.org/authors/?q=ai:hennion.benjaminSummary: We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic \(0\) field \(\boldsymbol{k}\), is the cyclic homology (over \(\boldsymbol{k})\). This equivalence is compatible with \(\lambda\)-operations. In particular, the relative algebraic \(K\)-theory functor fully determines the absolute cyclic homology over any field \(\boldsymbol{k}\) of characteristic \(0\). We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map \(\text{BGL}_\infty\rightarrow\text{K}\). The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory à la Lurie-Pridham.On the \(K\)-theoretic Hall algebra of a surfacehttps://www.zbmath.org/1475.190052022-01-14T13:23:02.489162Z"Zhao, Yu"https://www.zbmath.org/authors/?q=ai:zhao.yuSummary: In this paper, we define the \(K\)-theoretic Hall algebra for dimension \(0\) coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by \textit{A. Negut} [Sel. Math., New Ser. 25, No. 3, Paper No. 36, 57 p. (2019; Zbl 1427.14023)].Affine quiver Schur algebras and \(p\)-adic \(\mathrm{GL}_n\)https://www.zbmath.org/1475.200102022-01-14T13:23:02.489162Z"Miemietz, Vanessa"https://www.zbmath.org/authors/?q=ai:miemietz.vanessa"Stroppel, Catharina"https://www.zbmath.org/authors/?q=ai:stroppel.catharina-hThe (affine) Schur algebra is defined as the endomorphism algebra of certain permutation modules for the (affine) Iwahori-Matsumoto Hecke algebra, while the quiver Schur algebra attached to the cyclic quiver is defined as the \(\mathrm{GL}_\mathbf{d}\)-equivariant Borel-Moore homology of a ``Steinberg type'' variety equipped with the convolution product.
In the paper under review, an isomorphism between suitable completions of the above two algebras is established. This isomorphism endows the completed (affine) Schur algebra a grading, which is illustrated in the explicit example of \(\mathrm{GL}_2(\mathbb{Q}_5)\) in characteristic \(3\). The authors also provide some fundamental constructions for the (affine) Schur algebra, such as the generating sets, explicit faithful representations and geometrically adapted bases.
Reviewer: Li Luo (Shanghai)Character varieties for real formshttps://www.zbmath.org/1475.200122022-01-14T13:23:02.489162Z"Acosta, Miguel"https://www.zbmath.org/authors/?q=ai:acosta.miguelIn this article, the author gives a definition for character varieties for real forms of \(\mathrm{SL}(n,\mathbb{C})\). The main result is to characterize irreducible representations arising as conjugates of representations into real forms of \(\mathrm{SL}(n,\mathbb{C})\).
In particular, the author shows that the definition agrees with the topological definition given in the compact case \(\mathrm{SU}(n)\).
In the second section, the author recalls known facts about complex character varieties. In the third section, the author introduces the definition for character varieties of real form and proves the main result.
In the fourth section, the author uses his result to describe the real character variety of \(\mathbb{Z}/3*\mathbb{Z}/3\) into \(\mathrm{SU}(2,1)\) and \(\mathrm{SU}(3)\). The author uses the parametrization of \(\chi_{\mathrm{SL}(3,\mathbb{C})}(F_2)\) given by \textit{S. Lawton} [J. Algebra 313, No. 2, 782--801 (2007; Zbl 1119.13004)] and in particular find slices of representations parametrized by \textit{J. R. Parker} and \textit{P. Will} [Contemp. Math. 639, 327--348 (2015; Zbl 1360.20017)] and \textit{E. Falbel} et al. [Exp. Math. 25, No. 2, 219--235 (2016; Zbl 1353.57007)].
Reviewer: Clément Guérin (Esch-sur-Alzette)On the Humphreys conjecture on support varieties of tilting moduleshttps://www.zbmath.org/1475.200732022-01-14T13:23:02.489162Z"Achar, Pramod N."https://www.zbmath.org/authors/?q=ai:achar.pramod-n"Hardesty, William"https://www.zbmath.org/authors/?q=ai:hardesty.william-d"Riche, Simon"https://www.zbmath.org/authors/?q=ai:riche.simonLet \(G\) be a simply-connected semisimple algebraic group over an algebraically closed field \(\Bbbk\) of characteristic \(p > 0\), such that \(p\) is greater than the Coxeter number of \(G\). Let \(T \subset B \subset G\) be a fixed pair of a Borel subgroup in \(G\) and a maximal torus in \(B\), let \(\mathbf{X}\) denote the character lattice of \(T\) and let \(\mathbf{X}^+ \subset \mathbf{X}\) be the subset of dominant weights. Furthermore, let \(\dot{G}\) denote the Frobenius twist of \(G\) and let \(G_1\) be the kernel of the Frobenius morphism \(\mathrm{Fr}: G \rightarrow \dot{G}\). In the paper under review, the authors take for any \(\lambda \in \mathbf{X}^+\) the indecomposable tilting \(G\)-module \(\mathrm{T}(\lambda)\) of highest weight \(\lambda\) and consider the support variety \(V_{G_1}(\mathrm{T}(\lambda))\), which is a closed \(\dot{G}\)-stable subvariety of the nilpotent cone \(\mathcal{N}\) of \(\dot{G}\).
A conjecture of \textit{J. E. Humphreys}, stated in [AMS/IP Stud. Adv. Math. 4, 69--80 (1997; Zbl 0919.17013)] gives a conjectural description of the support varieties \(V_{G_1}(\mathrm{T}(\lambda))\) as closures of certain \(\dot{G}\)-orbits in \(\mathcal{N}\). This conjecture uses Lusztig's bijection between the set of \(\dot{G}\)-orbits in \(\mathcal{N}\) and the set of two-sided cells in the affine Weyl group of \(G\).
In an earlier work of the second author [Adv. Math. 329, 392--421 (2018; Zbl 1393.14043)], Humphreys conjecture was proved for \(G = \mathrm{SL}_n(\Bbbk)\) and \(p > n+1\) . The main result of the paper under review is to show that for any \(G\), the support variety \(V_{G_1}(\mathrm{T}(\lambda))\) always contains the variety predicted by Humphreys. Furthermore, there is an integer \(N > 0\) (depending only on the root system of \(G\)) such that if \(p > N\) the two varieties coincide (i.e., Humphreys conjecture is true). At the moment \(N\) cannot be determined explicitly, except for \(G = \mathrm{SL}_n(\Bbbk)\).
In addition, the authors also state and prove a variant of Humphreys conjecture involving ``relative support varieties''.
Reviewer: Elitza Hristova (Sofia)Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groupshttps://www.zbmath.org/1475.200752022-01-14T13:23:02.489162Z"Bardestani, Mohammad"https://www.zbmath.org/authors/?q=ai:bardestani.mohammad"Mallahi-Karai, Keivan"https://www.zbmath.org/authors/?q=ai:karai.keivan-mallahi"Salmasian, Hadi"https://www.zbmath.org/authors/?q=ai:salmasian.hadiSummary: Given a finite group $\text{G}$ and a field $K$, the \textit{faithful dimension} of $\text{G}$ over $K$ is defined to be the smallest integer $n$ such that $\text{G}$ embeds into $\mathrm{GL}_{n}(K)$. We address the problem of determining the faithful dimension of a $p$-group of the form $\mathscr{G}_{q}:=\exp (\mathfrak{g}\otimes_{\mathbb{Z}}\mathbb{F}_{q})$ associated to $\mathfrak{g}_{q}:=\mathfrak{g}\otimes_{\mathbb{Z}}\mathbb{F}_{q}$ in the Lazard correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra which is finitely generated as an abelian group. We show that in general the faithful dimension of $\mathscr{G}_{p}$ is a piecewise polynomial function of $p$ on a partition of primes into Frobenius sets. Furthermore, we prove that for $p$ sufficiently large, there exists a partition of $\mathbb{N}$ by sets from the Boolean algebra generated by arithmetic progressions, such that on each part the faithful dimension of $\mathscr{G}_{q}$ for $q:=p^{f}$ is equal to $fg(p^{f})$ for a polynomial $g(T)$. We show that for many naturally arising $p$-groups, including a vast class of groups defined by partial orders, the faithful dimension is given by a single formula of the latter form. The arguments rely on various tools from number theory, model theory, combinatorics and Lie theory.The Picard group of the forms of the affine line and of the additive grouphttps://www.zbmath.org/1475.200792022-01-14T13:23:02.489162Z"Achet, Raphaël"https://www.zbmath.org/authors/?q=ai:achet.raphaelSummary: We obtain an explicit upper bound on the torsion of the Picard group of the forms of \(\mathbb{A}_k^1\) and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of \(\mathbb{A}_k^1\) to be nontrivial and we give examples of nontrivial forms of \(\mathbb{A}_k^1\) with trivial Picard groups.Total positivity in Springer fibreshttps://www.zbmath.org/1475.200802022-01-14T13:23:02.489162Z"Lusztig, G."https://www.zbmath.org/authors/?q=ai:lusztig.georgeSummary: Let \(u\) be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at \(u\) with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.Loop of formal diffeomorphisms and Faà di Bruno coloop bialgebrahttps://www.zbmath.org/1475.200962022-01-14T13:23:02.489162Z"Frabetti, Alessandra"https://www.zbmath.org/authors/?q=ai:frabetti.alessandra"Shestakov, Ivan P."https://www.zbmath.org/authors/?q=ai:shestakov.ivan-pIn this paper, the authors study two generalizations of proalgebraic groups, on one side to representable functors on categories of non-commutative algebras, on the other side to functors taking divisions that is loops. The main motivation for the authors comes from two proalgebraic groups of formal series appearing in the renormalization in quantum field theory namely: The group of invertible series with constant term equal to 1, represented by the Hopf algebra of symmetric functions; and that of formal diffeomorphisms tangent to the identity represented by the Faà di Bruno Hopf algebra. The authors are interested in the relationship between the non-commutative algebras and sets of series. In Section 2.1 of Part 2, Loops and functors in loops, the authors describe coloops in an axiomatic way. In Section 2.2, Coloops in general categories, the authors give some easy examples of algebraic and non-algebraic loops on associative and non-associative algebras, and extensively study the loops of invertible series and that of formal diffeomorphisms. In Section 2.3, the authors study (Pro)algebraic loops. Part 3, Coloops of invertible and unitary elements. Section 3.1, Loop of invertible elements. In this section, the authors give an example of an abelian algebraic group which can be extended to associative algebras as a group, to alternative algebras as a loop, but not to non-associative algebras, even as a loop. In Section 3.2, the authors study loops of unitary elements. Section 3.3, Unitary Cayley-Dickson loops. In this section, they give an example of a loop which is not algebraic on associative algebras. Part 4, Coloops of invertible series. They give a definition of groups of invertible series \(\operatorname{Inv}(A)\), where \(A\) is a commutative algebra. And they show that the functors \(\operatorname{Inv}\) can be extended to non-associative algebras, as a proalgebraic loop. In Section 4.1, the authors study loops of invertible series. In Section 4.2, the authors study coloop bialgebras of invertible series. In Section 4.3, they study properties of the loop of invertible series. Part 5, Coloop of formal diffeomorphisms. Initially, they define the group of formal diffeomorphisms \(\operatorname{Diff}(A)\), where \(A\) is a commutative algebra. In Section 5.1, the authors study loops of formal diffeomorphisms, and define formal diffeomorphisms \(\operatorname{Diff}(A)\) in \(\lambda\) with coefficients in \(A\), where \(\lambda\) is a formal variable and \(A\) a unital associative algebra, non-necessarily commutative. In Section 5.2, the authors study Faà di Bruno coloop bialgebras and define it. In Section 5.3, they study Faà di Bruno co-operations in terms of recursive operators. In Section 5.4, the authors study functoriality of the diffeomorphism loops. In Section 5.5, they study properties of the diffeomorphism loops. In Appendix A, the authors presents categorical proofs with tangles. Tangle diagrams are an efficient tool to prove formal (categorical) properties. In the context of non-associative algebras, tangle diagrams have been used to code deformations of the enveloping algebra of a Malcev algebra, seen as the infinitesimal structure of a Moufang loop. In this Appendix, the authors present the list of the tangles needed to represent all the operations and the co-operations in coloops, with their defining identities.
Reviewer: C. Pereira da Silva (Curitiba)A topological realization of the congruence subgroup kernelhttps://www.zbmath.org/1475.220182022-01-14T13:23:02.489162Z"Scherk, John"https://www.zbmath.org/authors/?q=ai:scherk.johnSummary: Let \(\mathbf{G}\) be an almost simple, simply connected algebraic group defined over a number field \(k\), and let \(S\) be a finite set of places of kincluding all infinite places. The congruence subgroup kernel measures what proportion of \(S\)-arithmetic subgroups of \(\mathbf{G}\) are \(S\)-congruence subgroups. In this paper, a topological realization of the congruence subgroup kernel is given using the locally symmetric spaces associated with Gand its \(S\)-arithmetic subgroups. The construction uses the reductive Borel-Serre compactifications of these spaces. The congruence subgroup kernel then appears as a fundamental group.
For the entire collection see [Zbl 1403.11002].On depth zero L-packets for classical groupshttps://www.zbmath.org/1475.220262022-01-14T13:23:02.489162Z"Lust, Jaime"https://www.zbmath.org/authors/?q=ai:lust.jaime"Stevens, Shaun"https://www.zbmath.org/authors/?q=ai:stevens.shaunIn the paper under review, the authors study the structure of irreducible cuspidal representations of classical \(p\)-adic groups \(G\).
Let \(F\) be a nonarchimedean local field, \(\pi\) -- an irreducible cuspidal representation of a classical group \(G\) over \(F\), \(\mathcal A^\sigma(F)\) -- the set of equivalence classes of self-dual irreducible cuspidal representations \(\rho\) of the general linear group \(\mathrm{GL}_n(F)\). There is at most one real number \(s= s_\pi(\rho)\) such that the normalized parabolically induced representations \(\mathrm{Ind} \rho|\det(.)|^s_F \otimes \pi\) is reducible and then \(\rho\) is called a \textsl{reducible point} (Theorem 4.1). Denote the set of all reducible points by \(\mathrm{Red}(\pi) := \{(\rho,m) : \rho \in A^\sigma(F), m\in \mathbb N; 2s_\pi(\rho) = m+1 \}\). and the Jordan set by \(\mathrm{Jor}(\pi) := \{(\rho, m) : (\rho, m_\rho) \in \mathrm{Red}(\pi)\; \&\; m_\rho -m \in 2\mathbb Z_{\geq 0} \}\). A depth-zero reducible point \(\rho\) is called inertially reducible if it is in the lass of the twists \([\rho] = \{ \rho' = \rho\chi^2\}\), for some unramified character \(\chi\) of order \(2n_\rho\) and that \([\rho] \cap \mathcal A^\sigma(F) = \{ \varphi, \varphi' \}\). Denote the initial reducibility multiset by \(\mathrm{IRed}(\pi) = \{\{ ([\rho],m) : (\rho,m) \in \mathrm{Red}(\pi)\}\}\).
Let \(\varphi = \bigoplus_{\rho\in\mathrm{Jor}(\pi)} \varphi_\rho \otimes \mathrm{st}_m: \mathcal{W}_F \times \mathrm{SL}_2(\mathbb C) \to \hat{G}\rtimes \mathcal{W}_F\) be the Langlands parameter for \(G\) with the L-package \(\Pi_\varphi\) of \(\pi\), where \(\varphi_\rho\) is the irreducible representation of the Weil group \(\mathcal{W}_F\) corresponding to the representation \(\rho\) following the Langlands correspondence for general linear groups and \(\mathrm{st}_m\) is the \(m\)-dimensional irreducible representation of \(\mathrm{SL}_2(\mathbb C) \).
The main results of the paper are:
1. For a given tame Langlands parameter \(\varphi\), give an explicit description of the cuspidal irreducible representation \(\pi\) of a classical (specially, symplectic) group \(G\) such that \(\pi\) is in the union \(\Pi_\varphi \cup \Pi_{\varphi'}\) of L-packets of tame Langlands parameters. The algorithm to obtain the pair of Langlands parameters consists of:
\textbf{a.} To prove the equality \(\displaystyle\sum_{\sigma\in \mathcal A^\sigma(F)}\lfloor s_\pi(\rho))^2 \rfloor n_\rho \geq N_{\hat G}\), and therefore by using the opposite inequality by Moeglin, \(\displaystyle\sum_{\sigma\in \mathcal A^\sigma(F)}\lfloor s_\pi(\rho))^2 \rfloor n_\rho = N_{\hat G}\), where \(n_\rho\) is the unique natural number such that \(\rho\) is a representation of \(\mathrm{GL}_{n_\rho}\) and \(\displaystyle N_{\hat{G}}= \sum_{(\rho,m) \in \mathrm{Jor}(\pi) }mn_\rho\) is the dimension of the complex space on which the complex dual \(\hat{G}\) is acting on with multiplicity \(\displaystyle \sum_{(\rho,m) \in \mathrm{Jor}(\pi) }m\) of \(\rho\);
\textbf{b.} To give an explicit description of the set \(\mathrm{IRed}(\pi)\) of inertial reducible points;
\textbf{c.} To give an explicit description of the set of irreducible cuspidal representations \(\pi'\) of \(G\) with the equal multiset \(\mathrm{IRed}(\pi')=\mathrm{IRed}(\pi)\) in term of local data of \(\pi\).
2. For a depth-zero cuspidal irreducible representations \(\pi\) of a classical (specially, symplectic) group the author described the corresponding pair \(\{\varphi, \varphi'\}\) of tame Langlands parameters such that \(\pi \in \Pi_\varphi \cup \Pi_{\varphi'}\) of tame Langlands parameters (\S7).
Reviewer: Do Ngoc Diep (Hanoi)A family of \((p,n)\)-gonal Riemann surfaces with several \((p,n)\)-gonal groupshttps://www.zbmath.org/1475.301002022-01-14T13:23:02.489162Z"Reyes-Carocca, Sebastián"https://www.zbmath.org/authors/?q=ai:reyes-carocca.sebastianA compact Riemann surface \(S\) of genus at least 2 is said to be \((p, n)\)-gonal, where \(p\) is a prime, if there exists a group \(C\) of holomorphic automorphisms of \(S\) of order \(p\) -- which is called a \((p, n)\)-gonal group -- such that the orbit space \(S/C\) has genus \(n\) (where \(n \in \mathbb{N}\)). The classes of \((p, n)\)-gonal Riemann surfaces are generalisations of classes of compact Riemann surfaces that have been studied extensively in the past. For example the \((2,0)\)-gonal surfaces are precisely the hyperelliptic surfaces, and the \((2, n)\)-gonal surfaces are precisely the \(n\)-hyperelliptic surfaces. In this paper, it is shown that for any prime \(p \geq 3\) and for any \(n\in \mathbb{N}\) that is a multiple of \((p-1)\), there is a complex \(d\)-dimensional family \(\{S_{\alpha}\}\) of \((p, n)\)-gonal Riemann surfaces having the largest possible genus such that each \(S_{\alpha}\) admits more than one \((p, n)\)-gonal group, where \(d := 1+ n/(p-1)\). Here, the maximal genus equals \(2np + (p-1)^2\). It follows from the proof of this result that it also holds true for \(=2\) if \(n\) is odd.
Reviewer: Gautam Bharali (Bangalore)Every rational Hodge isometry between two K3 surfaces is algebraichttps://www.zbmath.org/1475.320082022-01-14T13:23:02.489162Z"Buskin, Nikolay"https://www.zbmath.org/authors/?q=ai:buskin.nikolaySummary: We present a proof that for any Hodge isometry \(\psi\colon H^2(S_1,{\mathbb{Q}}) \rightarrow H^2(S_2,{\mathbb{Q}})\) between any two Kähler \(K3\) surfaces \(S_1\) and \(S_2\) we can find a finite sequence of K3 surfaces and analytic \((2,2)\)-classes supported on successive products, such that the isometry \(\psi\) is the convolution of these classes. The proof of this fact implies that for projective \(S_1,S_2\) the class of \(\psi\) is algebraic. This proves a conjecture of \textit{I. R. Shafarevich} [Actes Congr. internat. Math. 1970, 1, 413--417 (1971; Zbl 0236.14016)].Koszul complexes and spectral sequences associated with Lie algebroidshttps://www.zbmath.org/1475.320112022-01-14T13:23:02.489162Z"Bruzzo, Ugo"https://www.zbmath.org/authors/?q=ai:bruzzo.ugo"Rubtsov, Vladimir N."https://www.zbmath.org/authors/?q=ai:rubtsov.vladimir-nSummary: We study some spectral sequences associated with a locally free \(\mathscr{O}_X\)-module \(\mathscr{A}\) which has a Lie algebroid structure. Here \(X\) is either a complex manifold or a regular scheme over an algebraically closed field \(k\). One spectral sequence can be associated with \(\mathscr{A}\) by choosing a global section \(V\) of \(\mathscr{A} \), and considering a Koszul complex with a differential given by inner product by \(V\). This spectral sequence is shown to degenerate at the second page by using Deligne's degeneracy criterion. Another spectral sequence we study arises when considering the Atiyah algebroid \(\mathscr{D}_{\mathscr{E}}\) of a holomolorphic vector bundle \(\mathscr{E}\) on a complex manifold. If \(V\) is a differential operator on \(\mathscr{E}\) with scalar symbol, i.e, a global section of \(\mathscr{D}_{\mathscr{E}} \), we associate with the pair \((\mathscr{E},V)\) a twisted Koszul complex. The first spectral sequence associated with this complex is known to degenerate at the first page in the untwisted \((\mathscr{E}=0)\) case.Notions of Stein spaces in non-Archimedean geometryhttps://www.zbmath.org/1475.320122022-01-14T13:23:02.489162Z"Maculan, Marco"https://www.zbmath.org/authors/?q=ai:maculan.marco"Poineau, Jérôme"https://www.zbmath.org/authors/?q=ai:poineau.jeromeAuthors' abstract: Let \(k\) be a non-Archimedean complete valued field and let \(X\) be a \(k\)-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: (1) for every complete valued extension \(k'\) of \(k\), every coherent sheaf on \(X\times_kk'\) is acyclic; (2) \(X\) is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex), and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, \(X\) is compact); (3) \(X\) admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. \par When \(X\) has no boundary the characterization is simpler: in (2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in (3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space.
Reviewer: Anatoly N. Kochubei (Kyïv)Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifoldshttps://www.zbmath.org/1475.320152022-01-14T13:23:02.489162Z"László, Tamás"https://www.zbmath.org/authors/?q=ai:laszlo.tamas"Nagy, János"https://www.zbmath.org/authors/?q=ai:nagy.janos"Némethi, András"https://www.zbmath.org/authors/?q=ai:nemethi.andrasThe authors use a symmetry of the topological zeta function and a reciprocity of polytopes to obtain a generalization of Seiberg-Witten invariants for some normal surface singularities. In particular this simplifies the computation of the (original) Seiberg-Witten invariants.
Take a normal surface singularity whose link, \(M\), is a rational homology sphere. The Seiberg-Witten invariant is computable from the resolution graph in a complicated way. First one builds the multi-variable topological zeta function \(\mathcal{Z}(\mathbf{t})\). The homology \(H:=H_1(M,\mathbb{Z})\) indexes the \(Spin^c\)-structure, and there is the natural decomposition \(\mathcal{Z}=\sum_{h\in H}\mathcal{Z}_h\). From the parts \(\{\mathcal{Z}_h\}\) one obtains certain quasi-polynomials. The value of each such quasi-polynomial at the origin is called the ``periodic constant \(pc(\mathcal{Z}_h)\) of \(\mathcal{Z}_h\)''. This \(pc(\mathcal{Z}_h)\) gives the Seiberg-Witten invariant corresponding to the \(h\)-spin structure.
The authors show that in fact \(pc(\mathcal{Z}_h)\) is an easy explicit sum of the coefficients of the ``dual'' series \(\mathcal{Z}_{[Z_K]-h}\), where \([Z_K]-h\in H\) is the Gorenstein-dual of \(h\).
Moreover, they establish the (unique, canonical) decomposition \(\mathcal{Z}_h=\mathcal{Z}_h^{neg}+P^+_h\). Here \(\mathcal{Z}^{neg}_h\) is a rational function ``with negative degree'', while \(P^+_h\) is a polynomial satisfying: \(P^+_h(1)=pc(\mathcal{Z}_h)\). Therefore \(P^+_h\) gives a multi-variable polynomial generalization of the (corresponding) Seiberg-Witten invariant. The authors determine \(P^+_h\) via the lattice points of some related polytopes. This is the topological analogue of the classical geometric genus formulae of Khovanskii and Morales.
Reviewer: Dmitry Kerner (Rehovot)Schubert decomposition for Milnor fibers of the varieties of singular matriceshttps://www.zbmath.org/1475.320172022-01-14T13:23:02.489162Z"Damon, James"https://www.zbmath.org/authors/?q=ai:damon.james-nThe variety of singular complex \(m\times m\) matrices is a hypersurface, defined by the determinant function, with nonisolated singularities. Its Milnor fibre at the origin is diffeomorphic to the global Milnor fibre \(F_m\) consisting of matrices with determinant 1, so \(F_m=\operatorname{SL}_m(\mathbb{C})\). A similar description holds for symmetric and skew-symmetric matrices, with \(m\) even and the Pfaffian as defining function in the last case. The Milnor fibres have as deformation retracts spaces which are symmetric spaces of classical type studied by Cartan: \(F_m=\operatorname{SL}_m(\mathbb{C})\) has \(\operatorname{SU}_m(\mathbb{C})\) as deformation retract. These compact models and results on the Schubert decomposition of Lie groups and symmetric spaces via the Cartan model are used to give cell decompositions of the global Milnor fibres. The Schubert decomposition is in terms of `unique ordered factorizations' of matrices as products of `pseudo-rotations'. In the case of symmetric or skewsymmetric matrices, this factorization has the form of iterated `Cartan conjugacies' by pseudo-rotations. For general matrices the duals of the Schubert cycles are represented as explicit monomials in the generators of the cohomology exterior algebra. The (skew) symmetric cases are also treated.
For a matrix singularity of any of the types the pull-backs of these cohomology classes generate a characteristic subalgebra of the cohomology of its Milnor fibre. Several problems on the cohomology of the Milnor fibre of matrix singularities are mentioned.
The final section indicates how the results extend to exceptional orbit hypersurfaces, complements and links.
Reviewer: Jan Stevens (Göteborg)Deformation and smoothing of singularitieshttps://www.zbmath.org/1475.320182022-01-14T13:23:02.489162Z"Greuel, Gert-Martin"https://www.zbmath.org/authors/?q=ai:greuel.gert-martinSummary: We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different invariants, such as the Milnor number, the Tjurina number, and the dimension of a smoothing component.
For the entire collection see [Zbl 1470.58001].Distinguished bases and monodromy of complex hypersurface singularitieshttps://www.zbmath.org/1475.320192022-01-14T13:23:02.489162Z"Ebeling, Wolfgang"https://www.zbmath.org/authors/?q=ai:ebeling.wolfgangSummary: We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of monodromy.
For the entire collection see [Zbl 1470.58001].The topology of the Milnor fibrationhttps://www.zbmath.org/1475.320202022-01-14T13:23:02.489162Z"Lê, Dũng Tráng"https://www.zbmath.org/authors/?q=ai:le-dung-trang."Nuño-Ballesteros, Juan José"https://www.zbmath.org/authors/?q=ai:nuno-ballesteros.juan-jose"Seade, José"https://www.zbmath.org/authors/?q=ai:seade.jose-aSummary: The fibration theorem for analytic maps near a critical point published by \textit{J. W. Milnor} [Singular points of complex hypersurfaces. Princeton University Press, Princeton, NJ (1968; Zbl 0184.48405)] is a cornerstone in singularity theory. It has opened several research fields and given rise to a vast literature. We review in this work some of the foundational results about this subject, and give proofs of several basic ``folklore theorems'' which either are not in the literature, or are difficult to find. Examples of these are that if two holomorphic map-germs are isomorphic, then their Milnor fibrations are equivalent, or that the Milnor number of a complex isolated hypersurface or complete intersection singularity \((X, \underline{0})\) does not depend on the choice of functions that define it. We glance at the use of polar varieties to studying the topology of singularities, which springs from ideas by René Thom. We give an elementary proof of a fundamental ``attaching-handles'' theorem, which is key for describing the topology of the Milnor fibers. This is also related to the so-called ``carousel'', that allows a deeper understanding of the topology of plane curves and has several applications in various settings. Finally we speak about Lê's conjecture concerning map-germs \(\mathbb{C}^2\to\mathbb{C}^3\), and about the Lê-Ramanujam theorem, which still is open in dimension 2.
For the entire collection see [Zbl 1470.58001].Finite-gap solutions of the Mikhalëv equationhttps://www.zbmath.org/1475.353072022-01-14T13:23:02.489162Z"Smirnov, A. O."https://www.zbmath.org/authors/?q=ai:smirnov.alexander-o"Pavlov, M. V."https://www.zbmath.org/authors/?q=ai:pavlov.maxim-v"Matveev, V. B."https://www.zbmath.org/authors/?q=ai:matveev.vladimir-b"Gerdjikov, V. S."https://www.zbmath.org/authors/?q=ai:gerdzhikov.vladimir-stefanovSummary: Two classes of multi-phase algebro-geometric solutions of Mikhalëv equation are constructed. The first class of solutions is associated with the Korteweg-de-Vries (KdV) equation. The second one is related to the solutions of the Kaup-Boussinesq (KB) equation. We have established interrelations among the multi-soliton, trigonometric, rational, elliptic and other known solutions of the KdV and KB equations and the solutions of Mikhalëv equations. We show that the number of linearly independent finite-gap solutions of Mikhalëv system is equal to the number of phases of these solutions. For each class of solutions we have constructed examples of explicit solutions of Mikhalëv equation. In the previous works cited below the solutions of the Mikhalëv system were described implicitly, being reduced to the solutions of appropriate Jacobi inversion problems. Here, to solve the Mikhalëv system explicitly, we used the formalism of Baker-Akhiezer functions.
For the entire collection see [Zbl 1461.37001].Poncelet property and quasi-periodicity of the integrable Boltzmann systemhttps://www.zbmath.org/1475.370622022-01-14T13:23:02.489162Z"Felder, Giovanni"https://www.zbmath.org/authors/?q=ai:felder.giovanniSummary: We study the motion of a particle in a plane subject to an attractive central force with inverse-square law on one side of a wall at which it is reflected elastically. This model is a special case of a class of systems considered by Boltzmann which was recently shown by \textit{G. Gallavotti} and \textit{I. Jauslin} [``A theorem on ellipses, an integrable system and a theorem of Boltzmann'', Preprint, \url{arXiv:2008.01955}] to admit a second integral of motion additionally to the energy. By recording the subsequent positions and momenta of the particle as it hits the wall, we obtain a three-dimensional discrete-time dynamical system. We show that this system has the Poncelet property: If for given generic values of the integrals one orbit is periodic, then all orbits for these values are periodic and have the same period. The reason for this is the same as in the case of the Poncelet theorem: The generic level set of the integrals of motion is an elliptic curve, and the Poincaré map is the composition of two involutions with fixed points and is thus the translation by a fixed element. Another consequence of our result is the proof of a conjecture of Gallavotti and Jauslin on the quasi-periodicity of the integrable Boltzmann system, implying the applicability of KAM perturbation theory to the Boltzmann system with weak centrifugal force.On intersections of polynomial semigroups orbits with plane lineshttps://www.zbmath.org/1475.371102022-01-14T13:23:02.489162Z"Mello, Jorge"https://www.zbmath.org/authors/?q=ai:mello.jorgeSummary: We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of \textit{D. Ghioca} et al. [Invent. Math. 171, No. 2, 463--483 (2008; Zbl 1191.14027)].A question for iterated Galois groups in arithmetic dynamicshttps://www.zbmath.org/1475.371112022-01-14T13:23:02.489162Z"Bridy, Andrew"https://www.zbmath.org/authors/?q=ai:bridy.andrew"Doyle, John R."https://www.zbmath.org/authors/?q=ai:doyle.john-r"Ghioca, Dragos"https://www.zbmath.org/authors/?q=ai:ghioca.dragos"Hsia, Liang-Chung"https://www.zbmath.org/authors/?q=ai:hsia.liang-chung"Tucker, Thomas J."https://www.zbmath.org/authors/?q=ai:tucker.thomas-jSummary: We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.The multidimensional truncated moment problem: Carathéodory numbers from Hilbert functionshttps://www.zbmath.org/1475.440062022-01-14T13:23:02.489162Z"di Dio, Philipp J."https://www.zbmath.org/authors/?q=ai:di-dio.philipp-j"Kummer, Mario"https://www.zbmath.org/authors/?q=ai:kummer.mario-denisThe paper under review is dedicated to some questions in the context of truncated moment problems, especially to the concept of Carathéodory number. We recall that the Carathéodory number is the minimal number \(N\) with the property that every moment sequence with a fixed truncation is the sum of \(N\) Dirac measures. The precise value of this number is not known in general but, nevertheless, evaluations in some important particular cases can be given. For the case of algebraic varieties with small gaps (that is, not all monomial are present), the bounds for the Carathéodory number, previously studied by other authors, are improved by the authors. In fact, they provide explicit lower lower and upper bounds on algebraic varieties, \(\mathbb R^n\), and \([0,1]^n\), also treating moment problems with small gaps. In particular, results concerning Hankel matrices are recaptured by the authors.
Reviewer: Florian-Horia Vasilescu (Villeneuve d'Ascq)The pentagrammum mysticum, twelve special conics and the twisted icosahedronhttps://www.zbmath.org/1475.510022022-01-14T13:23:02.489162Z"Le, Nguyen"https://www.zbmath.org/authors/?q=ai:le.nguyen-thi|le.nguyen-thinh|le.nguyen-thanh-tung"Wildberger, Norman J."https://www.zbmath.org/authors/?q=ai:wildberger.norman-johnThere is a huge amount of point and line configurations related to six points on a conic in the projective plane. The famous Pascal theorem is an origin of such a research. In the paper under review, the authors introduce pentagrammum mysticum as geometric and combinatorial structures related to five points in general position in the projective plane. The initial argument for such a kind of investigations is the fact that there is a unique conic passing through these five points. The discussed geometric objects are diagonal lines, exagonal points, diagonal points, exagonal lines, and dihedral conics. The authors show that the intersection data for twelve dihedral conics associated to the original five points is reflected in a distance transitive graph.
Reviewer: Georgi Hristov Georgiev (Shumen)Parabolic triangles, poles and centroid relationshttps://www.zbmath.org/1475.510152022-01-14T13:23:02.489162Z"Choi, Si Chun"https://www.zbmath.org/authors/?q=ai:choi.si-chun"Wildberger, N. J."https://www.zbmath.org/authors/?q=ai:wildberger.norman-johnConsider a triangle whose vertices \(A_1\), \(A_2\), \(A_3\) are on a given parabola in the Euclidean affine plane. The intersection points of the tangents to the parabola at \(A_1\), \(A_2\), \(A_3\) are the ``polar points'' \(X_1\), \(X_2\), \(X_3\). Combinations of these six points gives a bunch of new triangles. The authors investigate properties of the centroids of these triangles. For example, one of the many results states that the line through the centroids of \(A_1A_2A_3\) and \(X_1X_2X_3\) is parallel to the axis of the parabola (i.e. meets the parabola in its point at infinity); this was first indicated by D. Liu. The proofs of the theorems depend on explicit computations with respect to a suitable coordinate system. The authors conjecture that all results will be valid for conics which are tangent to the line of infinity.
Reviewer: Harald Löwe (Braunschweig)Initial steps in the classification of maximal mediated setshttps://www.zbmath.org/1475.520232022-01-14T13:23:02.489162Z"Hartzer, Jacob"https://www.zbmath.org/authors/?q=ai:hartzer.jacob"Röhrig, Olivia"https://www.zbmath.org/authors/?q=ai:rohrig.olivia"de Wolff, Timo"https://www.zbmath.org/authors/?q=ai:de-wolff.timo"Yürük, Oğuzhan"https://www.zbmath.org/authors/?q=ai:yuruk.oguzhanSummary: Maximal mediated sets (MMS), introduced by
\textit{B. Reznick} [Math. Ann. 283, No. 3, 431--464 (1989; Zbl 0637.10015)],
are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares.
In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices are isomorphic if and only if the simplices generate the same lattice up to permutations. Furthermore, we generalize a result of
\textit{S. Iliman} and the third author [Res. Math. Sci. 3, Paper No. 9, 35 p. (2016; Zbl 1415.11071)].
Practically, we fully characterize the MMS for all simplices of sufficiently small dimensions and maximal 1-norms. In particular, we experimentally prove a conjecture by Reznick for 2 dimensional simplices up to maximal 1-norm 150 and provide indications on the distribution of the density of MMS.Unobstructed symplectic packing by ellipsoids for tori and Hyperkähler manifoldshttps://www.zbmath.org/1475.530932022-01-14T13:23:02.489162Z"Entov, Michael"https://www.zbmath.org/authors/?q=ai:entov.michael"Verbitsky, Misha"https://www.zbmath.org/authors/?q=ai:verbitsky.mishaLet \((M,J,\omega)\) be a closed connected Kähler manifold. The complex structure \(J\) is called Campana-simple if the union of all complex, proper subvarieties of \(M\) has measure zero in \(M\). (The only other case is when this union is all \(M\).) Even-dimensional tori equipped with Kähler symplectic forms and closed hyper-Kähler manifolds of maximal holonomy are examples of Campana-simple manifolds. Furthermore \(J\) is said to be approximated by Campana-simple complex structures if \(J\) can be approximated by a smooth family \(\{J_t\}_{t\in B^{2n}\subset\mathbb{C}^n}\) of complex structures with \(J_0=J\) and if there is some sequence \({t_i}\subset B^{2n}\) with limit 0 such that every \(\{J_{t_i}\}\) is Campana-simple.
The main result of the present article asserts that if \(J\) can be approximated by Campana-simple complex structures then packing \(M\) by ellipsoids is unobstructed, i.e., any finite collection of pairwise disjoint closed ellipsoids in the standard symplectic \(\mathbb{R}^{2n}\) of total volume less than the symplectic volume of \(M\) can be symplectically embedded into \(M\). The proof of this flexibility result follows the sphere-packing results of \textit{D. McDuff} and \textit{L. Polterovich} [Invent. Math. 115, No. 3, 405--429 (1994; Zbl 0833.53028)] which study the structure of the symplectic cone in the cohomology of a blow-up of \(M\). The authors here consider instead the Kähler resolution (which is a smooth manifold) of the weighted blow-ups of \(M\) (which produce orbifolds). This result is a generalization of [\textit{J. Latschev} et al., Geom. Topol. 17, No. 5, 2813--2853 (2013; Zbl 1277.57024)].
Reviewer: Ferit Öztürk (Istanbul)Homological mirror symmetry of \(\mathbb{C} P^n\) and their products via Morse homotopyhttps://www.zbmath.org/1475.530952022-01-14T13:23:02.489162Z"Futaki, Masahiro"https://www.zbmath.org/authors/?q=ai:futaki.masahiro"Kajiura, Hiroshige"https://www.zbmath.org/authors/?q=ai:kajiura.hiroshigeSummary: We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category \(D^b( coh (X))\) of coherent sheaves on a toric manifold \(X\) is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. We instead consider the dual torus fibration \(\pi\): \(M \rightarrow B\) of the complement of the toric divisors in \(X\), where \(\overline{B}\) is the dual polytope of the toric manifold \(X\). A natural formulation of homological mirror symmetry in this setup is to define \(F u k(\overline{M})\), a variant of the Fukaya category, and show to the equivalence \(D^b(c o h(X)) \simeq D^b(F u k(\overline{M}))\). As an intermediate step, we construct the category \(Mo(P)\) of weighted Morse homotopy on \(P := \overline{B}\) as a natural generalization of the weighted Fukaya-Oh category proposed by \textit{M. Kontsevich} and \textit{Y. Soibelman} [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 203--263 (2001; Zbl 1072.14046)]. We then show that a full subcategory \(M o_{\mathcal{E}}(P)\) of \(Mo(P)\) generates \(D^b ( coh (X))\) for the cases \(X\) is a complex projective space and their products.
{\copyright 2021 American Institute of Physics}On homological mirror symmetry of toric Calabi-Yau threefoldshttps://www.zbmath.org/1475.530962022-01-14T13:23:02.489162Z"Gross, Mark"https://www.zbmath.org/authors/?q=ai:gross.mark.1|gross.mark.2|gross.mark-d"Matessi, Diego"https://www.zbmath.org/authors/?q=ai:matessi.diegoThe paper constructs Lagrangian sections and spheres on mirrors of toric Calabi-Yau manifolds, which are important ingredients in homological mirror symmetry. The authors propose an explicit correspondence between these Lagrangian sections and holomorphic line bundles on toric Calabi-Yau manifolds.
The celebrated SYZ program gives a geometric understanding of mirror symmetry by taking duals of Lagrangian fibrations. Lagrangian torus fibrations on toric Calabi-Yau manifolds \(\check{X}\) were constructed by the works of Goldstein and Gross, which are generalizations of the special Lagrangian fibration on \(\mathbb{C}^3\) constructed by Harvey-Lawson. Lagrangian fibrations on the mirrors \(X\) were constructed by Abouzaid-Auroux-Katzarkov. In general, the base of a Lagrangian fibration has a tropical affine structure. By taking the tropical limit, the projection of a Lagrangian submanifold to the base should give a tropical submanifold.
Away from singular fibers, a Lagrangian fibration is essentially given as \(T^*B/\Lambda^*\), where \(\Lambda^*\) is a family of lattices in \(T^*B\) over \(B\). The graph of a closed one-form on \(B\) gives a Lagrangian section. In the case of mirrors of toric Calabi-Yau manifolds \(\check{X}\), this can be encoded more combinatorially, namely it is constructed from a piecewise integral affine function supported on a polyhedral subdivision of the fan polytope. Note that such a combinatorial data also corresponds to a holomorphic line bundle over the toric Calabi-Yau manifolds \(X\).
This paper carries out these ideas in detail and successfully extends the sections over the whole SYZ base. It also constructs Lagrangian spheres in mirror toric Calabi-Yau manifolds \(\check{X}\), which correspond to tropical submanifolds supported in chambers of the base. These objects correspond to toric divisors in \(X\).
Reviewer: Siu Cheong Lau (Boston)\(C_2\)-equivariant stable homotopy from real motivic stable homotopyhttps://www.zbmath.org/1475.550112022-01-14T13:23:02.489162Z"Behrens, Mark"https://www.zbmath.org/authors/?q=ai:behrens.mark-joseph"Shah, Jay"https://www.zbmath.org/authors/?q=ai:shah.jayLet \(\mathrm{SH}(K)\) denote the category of motivic spectra over a field \(K\) in the sense of \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007)], and let \(\mathrm{Sp}\) denote the category of spectra. In the case that the base field \(K\) is the complex numbers \(\mathbb{C}\), Betti realization is a functor
\[
\mathrm{Be} \ \colon \mathrm{SH}(\mathbb{C}) \longrightarrow \mathrm{Sp},
\]
which sends a smooth scheme \(Z\) to its \(\mathbb{C}\)-points. In particular, there is an induced map
\[
\mathrm{Be} \ \colon \pi_{i,j}^{\mathbb{C}}(Z) = [S^{i-j} \wedge \mathbb{G}_m^j, X]_{\mathbb{C}} \longrightarrow \pi_{i} \mathrm{Be}(Z).
\]
At the prime \(2\) this map has been studied by \textit{D. Dugger} and \textit{D. C. Isaksen} [Geom. Topol. 14, No. 2, 967--1014 (2010; Zbl 1206.14041)], and has been studied at odd primes by \textit{S.-T. Stahn} [``The motivic Adams-Novikov spectral sequence at odd primes over $\mathbb{C}$ and $\mathbb{R}$'', Preprint, \url{arXiv:1606.06085}]. As a corollary of the results of the above mentioned authors, the authors of this work deduce that the category of \(p\)-complete spectra is equivalent to the category of \(p\)-complete cellular motivic spectra over \(\mathbb{C}\) after inverting the class \(\tau \in \pi_{0,-1}^\mathbb{C}(S^{0,0})^{\wedge}_p\) in the \(p\)-complete stable motivic homotopy groups of spheres. That is,
\[
\mathrm{Sp}^{\wedge}_p \cong \mathrm{SH}_{\mathrm{cell}}(\mathbb{C})^{\wedge}_p[\tau^{-1}].
\]
Working with the real numbers as the base field, \textit{T. Bachmann} [Compos. Math. 154, No. 5, 883--917 (2018; Zbl 06855294)] obtained an analogous result, namely that the category of spectra is equivalent (via Betti realization) to the stable real motivic homotopy category after inverting the class \(\rho \in \pi_{-1,-1}^\mathbb{R}(S^{0,0})\). That is,
\[
\mathrm{Sp} \cong \mathrm{SH}(\mathbb{R})[\rho^{-1}].
\]
The purpose of the paper under review is to prove results similar to those above, this time with respect to the \(C_2\)-Betti realization
\[
\mathrm{Be}^{C_2} \colon \mathrm{SH}(R) \longrightarrow \mathrm{Sp}^{C_2},
\]
where \(\mathrm{Sp}^{C_2}\) is the category of (genuine) \(C_2\)-spectra.
The method of approach chosen by the authors is via the isotropy seperation square of \textit{J. P. C. Greenlees} and \textit{J. P. May} [Generalized Tate cohomology. Providence, RI: American Mathematical Society (AMS) (1995; Zbl 0876.55003)], which expresses a \(C_2\)-spectrum \(Y\) as the homotopy pullback of its homotopy completion, \(Y^h = F((EC_2)_+, Y)\), its geometric localization \(Y^{\Phi} = Y \wedge \tilde{EC_2}\), and its equivariant Tate spectrum \(Y^t = (Y^h)^\Phi\).
Denote by \(\mathrm{Sp}^{hC_2}\) the full subcategory of \(C_2\)-spectra consisting of homotopically complete spectra, and let \(\mathrm{Sp}^{\Phi C_2}\) denote the full subcategory of geometrically local spectra. Since \(C_2\)-geometric fixed points yield an equivalence \(\mathrm{Sp}^{\Phi C_2} \cong \mathrm{Sp}\), Bachmann's theorem can be restated as an equivalence
\[
\mathrm{Sp}^{\Phi C_2} \cong \mathrm{SH}(\mathbb{R})[\rho^{-1}].
\]
It follows that the main task for the authors is to understand the homotopy completion of \(C_2\)-Betti realization. They prove that the right adjoint to \(p\)-complete, homotopy complete \(C_2\)-Betti realization
\[
\mathrm{Cell}\mathrm{Sing}^{C_2} \colon (S^{hC_2})^{\wedge}_p \longrightarrow \mathrm{SH}_{\mathrm{cell}}(\mathbb{R})^\wedge_p,
\]
is fully faithful.
In particular, these equivalences of categories and fully faithful functors give a method to compute all the required aspects of the isotropy separation square, in particular, the authors deduce a number of formulae, including
\[
\pi_{\ast, \ast}^{C_2}~ \widehat{\mathrm{Be}}_{p}^{C_2}(X)^h \cong \pi_{\ast, \ast}^\mathbb{R}~ X^\wedge_p[\tau^{-1}],
\]
and
\[
\pi_{\ast, \ast}^{C_2}~\widehat{\mathrm{Be}}_{p}^{C_2}(X)^t \cong \pi_{\ast,\ast}^\mathbb{R}~ X_p^\wedge[\tau^{-1}][\rho^{-1}].
\]
for \(X\) a \(p\)-complete cellular \(R\)-motivic spectrum, and
\[
\pi_{\ast, \ast}^{C_2}~ \mathrm{Be}^{C_2}(X)^\Phi \cong \pi_{\ast, \ast}^\mathbb{R} X [\rho^{-1}],
\]
for \(X\) an \(\mathbb{R}\)-motivic spectrum.
With this, the authors use the isotropy separation square to show that the \(p\)-complete \(C_2\)-equivariant stable homotopy category is a localization of the real motivic cellular stable homotopy category. In particular the authors carry out a number of computations of \(\pi_{\ast,\ast}^{C_2}~\widehat{\mathrm{Be}}^{C_2}_2 X\) for \(X = (H\mathbb{F}_2)_\mathbb{R}\), \((H\mathbb{Z}_2^\wedge)_\mathbb{R}\), and \(kgl_2^\wedge\).
Reviewer: Niall Taggart (Bonn)Thick ideals in equivariant and motivic stable homotopy categorieshttps://www.zbmath.org/1475.550232022-01-14T13:23:02.489162Z"Joachimi, Ruth"https://www.zbmath.org/authors/?q=ai:joachimi.ruthThis paper is an important advancement in the study of chromatic phenomena in the equivariant and motivic settings. The chromatic perspective has played a central role in stable homotopy theory over the past few decades, so the development of chromatic equivariant and motivic stable homotopy theory has become an important problem in algebraic topology.
The Hopkins-Smith Thick Subcategory Theorem [\textit{M. J. Hopkins} and \textit{J. H. Smith}, Ann. Math. (2) 148, No. 1, 1--49 (1998; Zbl 0924.55010)] implies that the thick ideals of the category of \(p\)-local finite spectra fit into an infinite chain, with each subcategory characterized by vanishing of a particular Morava K-theory. The main contribution of this article is the analysis of thick ideals in the categories of \(p\)-local finite equivariant and motivic spectra.
In the equivariant direction, Theorem 5 (attributed to Strickland) provides a lower bound on the lattice of thick ideals in the category of \(p\)-local finite \(G\)-spectra, where \(G\) is a finite group. More generally, Section 3 provides an account of Strickland's unpublished work discussing Morava K-theories, nilpotence, and thick ideals in the equivariant context. The specialization to the cyclic group of order two, \(C_2\), appears in Section 3.5.
In the motivic setting, the author works in the motivic stable homotopy category over a field \(k\) with a real or complex embedding. The main result is Theorem 13, which provides lower bounds on the lattices of thick tensor ideals in the categories of \(p\)-local finite \(k\)-motivic spectra. The restrictions on \(k\) are required to produce motivic thick ideals by pulling back the aforementioned nonequivariant (resp. \(C_2\)-equivariant) thick ideals along the Betti realization (resp. equivariant Betti realization) functors. The proof appears in Section 5.
In Sections 6 and 7, the author relates the thick ideals of Theorem 13 to thick ideals defined by the vanishing of motivic Morava K-theories. Proposition 78 implies that the thick ideal characterized by the vanishing of the \(n\)-th motivic Morava K-theory is contained in the thick ideal defined as the inverse image under Betti realization of the classical thick ideal characterized by vanishing of the analogous classical Morava K-theory. Corollary 84 uses the first motivic Hopf map, \(\eta\), to define a thick ideal which differs from all of the previously described thick ideals; in particular, this shows that the lattice of thick ideals in the motivic context is more complicated than its classical or \(C_2\)-equivariant counterparts.
Sections 8 and 9 explore related aspects of chromatic homotopy theory in the motivic context. Section 8 is concerned with type and the construction of type \(n\) motivic spectra, cf. Theorem 16, and Section 9 is concerned with nilpotence and Bousfield classes. The main results appear in Sections 9.4 and 9.5, which relate the Bousfield classes of various motivic spectra. Many of the results in Section 9 restrict to primes \(p>2\), which simplifies the necessary analysis using the motivic Steenrod algebra and motivic Adams spectral sequence.
There have been several closely related advances in chromatic equivariant and motivic stable homotopy theory since the paper under review was written. On the equivariant side, \textit{P. Balmer} and \textit{B. Sanders} [Invent. Math. 208, No. 1, 283--326 (2017; Zbl 1373.18016)] and \textit{T. Barthel} et al. [ibid. 216, No. 1, 215--240 (2019; Zbl 1417.55016)] have completely determined the lattice of prime thick tensor ideals. On the motivic side, \textit{S.-T. Stahn} [Stable motivic homotopy groups and periodic self maps at odd primes. Wuppertal: Bergische Universität Wuppertal (PhD Thesis) (2019), \url{urn:nbn:de:hbz:468-20190130-102816-9}] has shown that the two thick subcategories constructed in the paper under review using Betti realization and Morava K-theory are actually different, which was left open in this work. \textit{A. Krause} [Periodicity in motivic homotopy theory and over \(\mathrm{BP}*\mathrm{BP}\). Bonn: Rheinische Friedrich-Wilhelms-Universität Bonn (PhD Thesis) (2018), \url{https://nbn-resolving.org/urn:nbn:de:hbz:5n-51245}] has defined motivic Morava K-theories which detect more exotic forms of periodicity over the complex numbers.
For the entire collection see [Zbl 1460.55001].
Reviewer: James D. Quigley (Ithaca)Fractional Dehn twists and modular invariantshttps://www.zbmath.org/1475.570342022-01-14T13:23:02.489162Z"Liu, Xiao-Lei"https://www.zbmath.org/authors/?q=ai:liu.xiaoleiSummary: In this paper, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some uniform lower bounds of non-zero fractional Dehn twist coefficients.The gamma filtrations of \(K\)-theory of complete flag varietieshttps://www.zbmath.org/1475.570482022-01-14T13:23:02.489162Z"Yagita, Nobuaki"https://www.zbmath.org/authors/?q=ai:yagita.nobuakiFrom the introduction: Let \(p\) be a prime number. Let \({K^*}_{top}(X)\) be the complex \(K\)-theory localized at \(p\) for a topological space \(X\). There are two typical filtrations for \({K^0}_{top}(X)\), the topological filtration defined by \textit{M. F. Atiyah} [Publ. Math., Inst. Hautes Étud. Sci. 9, 247--288 (1961; Zbl 0107.02303)] and the \(\gamma\)-filtration defined by Grothendieck. Let us write by \({gr^*}_{top}(X)\) and \({gr^*}_{\gamma}(X)\) the associated graded rings for these filtrations. Let \(G\) and \(T\) be a connected compact Lie group and its maximal torus. Then \({gr^*}_{top}(G/T)\simeq {H^*}(G/T)_{(p)}\). However when \({H^*}(G)\) has \(p\)-torsion, it is not isomorphic to \({gr^*}_{\gamma}(G/T)\). In this paper, we try to compute \({gr^*}_{\gamma}(G/T)\) using the Chow rings of corresponding versal flag varieties.
Reviewer: Cenap Özel (İzmir)Noncommutative symplectic manifolds for self-injective Nakayama algebrashttps://www.zbmath.org/1475.580052022-01-14T13:23:02.489162Z"Pogorzały, Zygmunt"https://www.zbmath.org/authors/?q=ai:pogorzaly.zygmuntSummary: In the paper, we study noncommutative symplectic manifolds \((A,[\![\omega]\!])\), where \(A\) is a self-injective Nakayama algebra. There is given a full description of exact symplectic manifolds for \(A\) being the trivial extension of an hereditary Nakayama algebra.Graph Riemann hypothesis and Ihara zeta function of nonregular Ramanujan graph generated by \(p\)-adic chaoshttps://www.zbmath.org/1475.600152022-01-14T13:23:02.489162Z"Naito, Koichiro"https://www.zbmath.org/authors/?q=ai:naito.koichiroSummary: In our previous papers, applying chaotic properties of the \(p\)-adic dynamical system given by the \(p\)-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the sequences of these pseudorandom numbers given by this generator, we construct some pseudorandom adjacency matrices and their graphs. Since the regular Ramanujan graph satisfies the graph Riemann hypothesis, we numerically investigate our pseudorandom nonregular graphs by calculating the distributions of poles of the Ihara zeta functions, which are obtained by substituting our pseudorandom adjacency matrices into the Ihara determinant formula.Autocovariance varieties of moving average random fieldshttps://www.zbmath.org/1475.622892022-01-14T13:23:02.489162Z"Améndola, Carlos"https://www.zbmath.org/authors/?q=ai:amendola.carlos"Pham, Viet Son"https://www.zbmath.org/authors/?q=ai:pham.viet-sonSummary: We study the autocovariance functions of moving average random fields over the integer lattice \(\mathbb{Z}^d\) from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.On variational principle and canonical structure of gravitational theory in double-foliation formalismhttps://www.zbmath.org/1475.830052022-01-14T13:23:02.489162Z"Aghapour, Sajad"https://www.zbmath.org/authors/?q=ai:aghapour.sajad"Jafari, Ghadir"https://www.zbmath.org/authors/?q=ai:jafari.ghadir"Golshani, Mehdi"https://www.zbmath.org/authors/?q=ai:golshani.mehdiCompact maximal hypersurfaces in globally hyperbolic spacetimeshttps://www.zbmath.org/1475.830062022-01-14T13:23:02.489162Z"Aledo, Juan A."https://www.zbmath.org/authors/?q=ai:aledo.juan-angel"Rubio, Rafael M."https://www.zbmath.org/authors/?q=ai:rubio.rafael-maria"Salamanca, Juan J."https://www.zbmath.org/authors/?q=ai:salamanca.juan-jesusTowards a swampland global symmetry conjecture using weak gravityhttps://www.zbmath.org/1475.830302022-01-14T13:23:02.489162Z"Daus, Tristan"https://www.zbmath.org/authors/?q=ai:daus.tristan"Hebecker, Arthur"https://www.zbmath.org/authors/?q=ai:hebecker.arthur"Leonhardt, Sascha"https://www.zbmath.org/authors/?q=ai:leonhardt.sascha"March-Russell, John"https://www.zbmath.org/authors/?q=ai:march-russell.johnSummary: It is widely believed and in part established that exact global symmetries are inconsistent with quantum gravity. One then expects that approximate global symmetries can be \textit{quantitatively} constrained by quantum gravity or swampland arguments. We provide such a bound for an important class of global symmetries: Those arising from a gauged \(U(1)\) with the vector made massive via Higgsing with an axion. The latter necessarily couples to instantons, and their action can be constrained, using both the electric and magnetic version of the axionic weak gravity conjecture, in terms of the cutoff of the theory. As a result, instanton-induced symmetry breaking operators with a suppression factor not smaller than \(\exp(- M_{\mathrm{P}}^2/\Lambda^2)\) are present, where \(\Lambda\) is a cutoff of the 4d effective theory. We provide a general argument and clarify the meaning of \(\Lambda \). Simple 4d and 5d models are presented to illustrate this, and we recall that this is the standard way in which things work out in string compactifications with brane instantons. The relation of our constraint to bounds that can be derived from wormholes or gravitational instantons and to those motivated by black-hole effects at finite temperature are discussed, and we present a generalization of the Giddings-Strominger wormhole solution to the case of a gauge-derived \(U(1)\) global symmetry. Finally, we discuss potential loopholes to our arguments.On the symmetry of \(T \bar{T}\) deformed CFThttps://www.zbmath.org/1475.830442022-01-14T13:23:02.489162Z"He, Miao"https://www.zbmath.org/authors/?q=ai:he.miao"Gao, Yi-hong"https://www.zbmath.org/authors/?q=ai:gao.yihongSummary: We propose a symmetry of \(T \bar{T}\) deformed 2D CFT, which preserves the trace relation. The deformed conformal killing equation is obtained. Once we consider the background metric runs with the deformation parameter \(\mu \), the deformation contributes an additional term in conformal killing equation, which plays the role of renormalization group flow of metric. The conformal symmetry coincides with the fixed point. On the gravity side, this deformed conformal killing equation can be described by a new boundary condition of \(\mathrm{AdS}_3\). In addition, based on the deformed conformal killing equation, we derive that the stress tensor of the deformed CFT equals to Brown-York's quasilocal stress tensor on a finite boundary with a counterterm. For a specific example, BTZ black hole, we get \(T \bar{T}\) deformed conformal killing vectors and the associated conserved charges are also studied.Geometry and symmetries of null G-structureshttps://www.zbmath.org/1475.830862022-01-14T13:23:02.489162Z"Papadopoulos, G."https://www.zbmath.org/authors/?q=ai:papadopoulos.george-d|papadopoulos.george.1|papadopoulos.george-k.1|papadopoulos.g-j|papadopoulos.g-th|papadopoulos.g-j.1|papadopoulos.georgios-o|papadopoulos.george-a.1|papadopoulos.george-k|papadopoulos.george-a|papadopoulos.george|papadopoulos.gregoryStudy of semiclassical instability of the Schwarzschild AdS black hole in the large \(D\) limithttps://www.zbmath.org/1475.830902022-01-14T13:23:02.489162Z"Sadhu, Amruta"https://www.zbmath.org/authors/?q=ai:sadhu.amruta"Suneeta, Vardarajan"https://www.zbmath.org/authors/?q=ai:suneeta.vardarajanJT gravity and the asymptotic Weil-Petersson volumehttps://www.zbmath.org/1475.831082022-01-14T13:23:02.489162Z"Kimura, Yusuke"https://www.zbmath.org/authors/?q=ai:kimura.yusukeSummary: A path integral in Jackiw-Teitelboim (JT) gravity is given by integrating over the volume of the moduli of Riemann surfaces with boundaries, known as the ``Weil-Petersson volume,'' together with integrals over wiggles along the boundaries. The exact computation of the Weil-Petersson volume \(V_{g, n}(b_1, \dots, b_n)\) is difficult when the genus \(g\) becomes large. We utilize two partial differential equations known to hold on the Weil-Petersson volumes to estimate asymptotic behaviors of the volume with two boundaries \(V_{g, 2}(b_1, b_2)\) and the volume with three boundaries \(V_{g, 3}(b_1, b_2, b_3)\) when the genus \(g\) is large. Furthermore, we present a conjecture on the asymptotic expression for general \(V_{g, n}(b_1, \dots, b_n)\) with \(n\) boundaries when \(g\) is large.On the security properties of Russian standardized elliptic curveshttps://www.zbmath.org/1475.940932022-01-14T13:23:02.489162Z"Alekseev, E. K."https://www.zbmath.org/authors/?q=ai:alekseev.evgeny-k"Nikolaev, V. D."https://www.zbmath.org/authors/?q=ai:nikolaev.v-d"Smyshlyaev, S. V."https://www.zbmath.org/authors/?q=ai:smyshlyaev.stanislav-vSummary: In the last two decades elliptic curves have become a necessary part of numerous cryptographic primitives and protocols. Hence it is extremely important to use the elliptic curves that do not weaken the security of such protocols. We investigate the elliptic curves used with GOST R 34.10-2001, GOST R 34.10-2012 and the accompanying algorithms, their security properties and generation process.Constructions of elliptic curves endomorphismshttps://www.zbmath.org/1475.941412022-01-14T13:23:02.489162Z"Nesterenko, A. Yu."https://www.zbmath.org/authors/?q=ai:nesterenko.aleksey-yuSummary: Let \(\mathbb{K}\) be an imaginary quadratic field. Consider an elliptic curve \(E(\mathbb{F}_p)\) defined over prime field \(\mathbb{F}_p\) with given ring of endomorphisms \(o_{\mathbb{K}}\), where \(o_{\mathbb{K}}\) is an order in a ring of integers \(\mathbb{Z}_{\mathbb{K}}\).
An algorithm permitting to construct endomorphism of the curve \(E(\mathbb{F}_p)\) corresponding to the complex number \(\tau\in o_{\mathbb{K}}\) is presented. The endomorphism is represented as a pair of rational functions with coefficients in \(\mathbb{F}_p\). To construct these functions we use continued fraction expansion for values of Weierstrass function. After that we reduce the rational functions modulo prime ideal in finite extension of \(\mathbb{K}\). One can use such endomorphism for elliptic curve point exponentiation.On stopping sets of AG codes over certain curves with separated variableshttps://www.zbmath.org/1475.942042022-01-14T13:23:02.489162Z"Tenório, Wanderson"https://www.zbmath.org/authors/?q=ai:tenorio.wanderson"Tizziotti, Guilherme Chaud"https://www.zbmath.org/authors/?q=ai:tizziotti.guilherme-cEditorial remark: No review copy delivered.