Recent zbMATH articles in MSC 14Hhttps://www.zbmath.org/atom/cc/14H2021-04-16T16:22:00+00:00WerkzeugIntegrable systems and algebraic geometry. A celebration of Emma Previato's 65th birthday. Volume 1.https://www.zbmath.org/1456.140032021-04-16T16:22:00+00:00"Donagi, Ron (ed.)"https://www.zbmath.org/authors/?q=ai:donagi.ron-y"Shaska, Tony (ed.)"https://www.zbmath.org/authors/?q=ai:shaska.tanushPublisher's description: Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painlevé and many other notable special equations.
The articles of this volume will be reviewed individually. For Vol. 2 see [Zbl 1456.14004].On a type of permutation rational functions over finite fields.https://www.zbmath.org/1456.112192021-04-16T16:22:00+00:00"Hou, Xiang-dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dong"Sze, Christopher"https://www.zbmath.org/authors/?q=ai:sze.christopherSummary: Let \(p\) be a prime and \(n\) be a positive integer. Let \(f_b(X)=X+(X^p-X+b)^{-1}\), where \(b\in\mathbb{F}_{p^n}\) is such that \(\text{Tr}_{p^n/p}(b)\neq 0\). In 2008, \textit{J. Yuan} et al. [Finite Fields Appl. 14, No. 2, 482--493 (2008; Zbl 1211.11136)] showed that for \(p=2,3,f_b\) permutes \(\mathbb{F}_{p^n}\) for all \(n\geq 1\). Using the Hasse-Weil bound, we show that when \(p>3\) and \(n\geq 5, f_b\) does not permute \(\mathbb{F}_{p^n} \). For \(p>3\) and \(n=2\), we prove that \(f_b\) permutes \(\mathbb{F}_{p^2}\) if and only if \(\text{Tr}_{p^2/p}(b)=\pm 1\). We conjecture that for \(p>3\) and \(n=3,4,f_b\) does not permute \(\mathbb{F}_{p^n}\).Equations and rational points of the modular curves \(X_0^+(p)\).https://www.zbmath.org/1456.111112021-04-16T16:22:00+00:00"Mercuri, Pietro"https://www.zbmath.org/authors/?q=ai:mercuri.pietroSummary: Let \(p\) be an odd prime number and let \(X_0^+(p)\) be the quotient of the classical modular curve \(X_0(p)\) by the action of the Atkin-Lehner operator \(w_p\). In this paper, we show how to compute explicit equations for the canonical model of \(X_0^+(p)\). Then we show how to compute the modular parametrization, when it exists, from \(X_0^+(p)\) to an isogeny factor \(E\) of dimension 1 of its Jacobian \(J_0^+(p)\). Finally, we show how to use this map to determine the rational points on \(X_0^+(p)\) up to a large fixed height.The integral Chow ring of \(\overline{M}_2\).https://www.zbmath.org/1456.140352021-04-16T16:22:00+00:00"Larson, Eric"https://www.zbmath.org/authors/?q=ai:larson.ericThe author computes the Chow ring of the moduli stack \(\overline{M}_2\) of stable curves of genus 2 with integral coefficients.
More precisely, the author gives a full description of the integral Chow ring, over any field of characteristic distinct from 2 and 3, in terms of Chern classes of the Hodge bundle and a class of the boundary stratum \(\Delta_1\) of curves with a disconnecting node.
This is done by considering a stratification of \(\overline{M}_2\) into \(\Delta_1\) and its complement \(\overline{M}_2\backslash\Delta_1:\) each of these strata is proved to be isomorphic to quotients of open subsets in affine spaces by linear algebraic groups and their Chow rings have been computed using equivariant intersection theory. Then, the author proves the injectivity of\[\operatorname{CH}^*(\Delta_1)\rightarrow \operatorname{CH}^{*+1}(\overline{M}_2)\] and the non vanishing of a finite number of classes in \(\operatorname{CH}^*(\overline{M}_2)\otimes\mathbb{Z}/2\mathbb{Z}\), by computing the Chow ring of the universal family of bielliptic curves of genus 2.
This, together with the Chow ring of the two strata, gives the full Chow ring of \(\overline{M}_2\).
Reviewer: Angelina Zheng (Padova)New examples of maximal curves with low genus.https://www.zbmath.org/1456.111152021-04-16T16:22:00+00:00"Bartoli, Daniele"https://www.zbmath.org/authors/?q=ai:bartoli.daniele"Giulietti, Massimo"https://www.zbmath.org/authors/?q=ai:giulietti.massimo"Kawakita, Motoko"https://www.zbmath.org/authors/?q=ai:kawakita.motoko-qiu"Montanucci, Maria"https://www.zbmath.org/authors/?q=ai:montanucci.mariaSummary: In this paper, explicit equations for algebraic curves with genus 4, 5, and 10 already studied in characteristic zero, are analyzed in positive characteristic \(p\). We show that these curves have an interesting behaviour on the number of their rational places. Namely, they are either maximal or minimal over the finite field with \(p^2\) elements for infinitely many \(p\)'s. The key tool is the investigation of their Jacobian decomposition. Lists of small \(p\)'s for which maximality holds are provided. In some cases we also describe the automorphism group of the curve.Multi-point codes from the GGS curves.https://www.zbmath.org/1456.941362021-04-16T16:22:00+00:00"Hu, Chuangqiang"https://www.zbmath.org/authors/?q=ai:hu.chuangqiang"Yang, Shudi"https://www.zbmath.org/authors/?q=ai:yang.shudiAlgebraic curves over finite fields can be used to obtain error correcting codes since the seminal work of Goppa in the early 1980s. Algebraic geometric (AG) codes have ``good'' parameters when the underlying curve has many rational points with respect to its genus. For this reason, maximal curves (i.e. curves attaining the upper bound in the Hasse-Weil bound) have been widely investigated. Recently, AG codes from Hermitian, Suzuki, Klein quartic, GK, and GGS curves and their quotients attracted a lot of attention. Most of the constructions of AG codes are one-point. In the case of multi-point AG codes, the main problem is a suitable description of Riemann-Roch spaces associated with divisors having a large support.
This paper deals with the construction of AG codes defined from GGS curves, a generalization of the GK curve. In particular, the authors describe bases for the Riemann-Roch spaces associated with some rational places, and characterize explicitly the Weierstrass semigroups and pure gaps (a generalization of gaps) by an exhaustive computation for the basis of Riemann-Roch spaces from GGS curves. As a byproduct, multi-point codes with parameters achieving new records are obtained.
Reviewer: Daniele Bartoli (Perugia)Introduction to Lipschitz geometry of singularities. Lecture notes of the international school on singularity theory and Lipschitz geometry, Cuernavaca, Mexico, June 11--22, 2018.https://www.zbmath.org/1456.580022021-04-16T16:22:00+00:00"Neumann, Walter (ed.)"https://www.zbmath.org/authors/?q=ai:neumann.walter-d"Pichon, Anne (ed.)"https://www.zbmath.org/authors/?q=ai:pichon.annePublisher's description: This book presents a broad overview of the important recent progress which led to the emergence of new ideas in Lipschitz geometry and singularities, and started to build bridges to several major areas of singularity theory. Providing all the necessary background in a series of introductory lectures, it also contains Pham and Teissier's previously unpublished pioneering work on the Lipschitz classification of germs of plane complex algebraic curves.
While a real or complex algebraic variety is topologically locally conical, it is in general not metrically conical; there are parts of its link with non-trivial topology which shrink faster than linearly when approaching the special point. The essence of the Lipschitz geometry of singularities is captured by the problem of building classifications of the germs up to local bi-Lipschitz homeomorphism. The Lipschitz geometry of a singular space germ is then its equivalence class in this category.
The book is aimed at graduate students and researchers from other fields of geometry who are interested in studying the multiple open questions offered by this new subject.
The articles of this volume will be reviewed individually.
Indexed articles:
\textit{Aguilar-Cabrera, Haydée; Cisneros-Molina, José Luis}, Geometric viewpoint of Milnor's fibration theorem, 1-43 [Zbl 07303679]
\textit{Snoussi, Jawad}, A quick trip into local singularities of complex curves and surfaces, 45-71 [Zbl 07303680]
\textit{Neumann, Walter D.}, 3-manifolds and links of singularities, 73-86 [Zbl 07303681]
\textit{Trotman, David}, Stratifications, equisingularity and triangulation, 87-110 [Zbl 07303682]
\textit{Ruas, Maria Aparecida Soares}, Basics on Lipschitz geometry, 111-155 [Zbl 07303683]
\textit{Birbrair, Lev; Gabrielov, Andrei}, Surface singularities in \(\mathbb{R}^4\) : first steps towards Lipschitz knot theory, 157-166 [Zbl 07303684]
\textit{Pichon, Anne}, An introduction to Lipschitz geometry of complex singularities, 167-216 [Zbl 07303685]
\textit{Giles Flores, Arturo; da Silva, Otoniel Nogueira; Teissier, Bernard}, The biLipschitz geometry of complex curves: an algebraic approach, 217-271 [Zbl 07303686]
\textit{Popescu-Pampu, Patrick}, Ultrametrics and surface singularities, 273-308 [Zbl 07303687]
\textit{Pham, Frédéric; Teissier, Bernard}, Lipschitz fractions of a complex analytic algebra and Zariski saturation, 309-337 [Zbl 07303688]Nef cone and Seshadri constants on products of projective bundles over curves.https://www.zbmath.org/1456.140532021-04-16T16:22:00+00:00"Karmakar, Rupam"https://www.zbmath.org/authors/?q=ai:karmakar.rupam"Misra, Snehajit"https://www.zbmath.org/authors/?q=ai:misra.snehajitIn the paper under review the authors study the geometry of the nef cone and the Seshadri constants on products of projective bundles over curves. Let \(E_{1}\) and \(E_{2}\) be two vector bundles over a smooth curve \(C\) of rank \(r_{1}\) and \(r_{2}\), respectively, and degree \(d_{1}\) and \(d_{2}\). Let \(\mathbb{P}(E_{i}) =\mathrm{Proj}(\bigoplus_{d\geq 0} \mathrm{Sym}^{d} (E_{i}))\). Consider the fiber product \(X = \mathbb{P}(E_{1}) \times_{C} \mathbb{P}(E_{2})\) over \(C\) and define \(p_{i} : X \rightarrow \mathbb{P}(E_{i})\) for \(i \in \{1,2\}\). Denote by \(\eta_{i} = [\mathcal{O}_{\mathbb{P}(E_{i})}(1)] \in N^{1}(\mathbb{P}(E_{i}))\). The first result of the paper provides us a full description of the nef cone of \(X\).
Theorem A. Let \(E_{1}, E_{2}\) be two vector bundles on a smooth complex curve \(C\), then
\[\mathrm{Net}(X) = \{ a\tau_{1} + b\tau_{2} + cF \, : \, a,b,c \in \mathbb{R}_{\geq 0}\},\]
where \(\tau_{1} =p_{2}^{*} \eta_{1} - \mu_{11}F\) and \(\tau_{2} = p_{1}^{*} \eta_{2} - \mu_{21}F\) and \(F\) is nef, where \(\mu_{11}, \mu_{21}\) are the smallest slopes of any torsion-free quotients of \(E_{1}\) and \(E_{2}\), respectively (in the sense of the Harder-Narasimhan filtration).
Let us recall that if \(X\) is a smooth complex projective variety and \(L\) a nef line bundle on \(X\), then the Seshadri constant of \(L\) at a point \(x \in X\) is defined as
\[\varepsilon(X,L;x) = \mathrm{inf}_{C \subset X} \bigg\{ \frac{L \cdot C}{\mathrm{mult}_{x} (C)}\bigg\},\]
where the infimum is taken over all irreducible curves in \(X\) passing through \(X\) having multiplicity \(\mathrm{mult}_{x}(C)\) at \(X\).
Theorem B. Let \(E_{1}, E_{2}\) be two vector bundles on a smooth curve \(C\) with \(\mu_{11}, \mu_{21}\) being the smallest slopes of any torsion-free quotient of \(E_{1}\) and \(E_{2}\), respectively. Let \(L\) be an ample line bundle on \(X\) which is numerically equivalent to \(a \tau_{1} + b\tau_{2} + cF \in N^{1}(X)\). Then the Seshadri constants of \(L\) satisfy, for any \(x \in X\), the following inequality
\[\varepsilon(X,L;x) \geq\min\{a,b,c\}.\]
Moreover, if \(a = \min \{a,b,c\}\), then \(\varepsilon(X,L;x) = a\) for any \(x \in X\), or if \(b = \min\{a,b,c\}\), then \(\varepsilon(X,L;x) = b\) for any \(x \in X\).
Reviewer: Piotr Pokora (Kraków)Integrable systems and algebraic geometry. A celebration of Emma Previato's 65th birthday. Volume 2.https://www.zbmath.org/1456.140042021-04-16T16:22:00+00:00"Donagi, Ron (ed.)"https://www.zbmath.org/authors/?q=ai:donagi.ron-y"Shaska, Tony (ed.)"https://www.zbmath.org/authors/?q=ai:shaska.tanushPublisher's description: Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. The articles in this second volume discuss areas related to algebraic geometry, emphasizing the connections of this central subject to integrable systems, arithmetic geometry, Riemann surfaces, coding theory and lattice theory.
The articles of this volume will be reviewed individually. For Vol. 1 see [Zbl 1456.14003].Intersection cohomology of pure sheaf spaces using Kirwan's desingularization.https://www.zbmath.org/1456.140152021-04-16T16:22:00+00:00"Chung, Kiryong"https://www.zbmath.org/authors/?q=ai:chung.kiryong"Yoon, Youngho"https://www.zbmath.org/authors/?q=ai:yoon.younghoLet \(\mathbf{M}_n\) be the space parametrizing semi-stable sheaves \(F\) on \(\mathbb {P}^n\) with a linear resolution \[0\to\mathcal {O}_{\mathbb {P}^n}(-1)^2 \to \mathcal {O}_{\mathbb {P}^n}^2\to F\to 0.\] \(\mathbb {M}_n\) is an integral normal variety, \(\dim \mathbb {M}_n =4n-3\), which is the Simpsons compactification of twisted sheaves \(\mathcal{I}_{L,Q}(1)\), where \(Q\subset \mathbb {P}^n\) is a rank \(4\) hyperquadric and \(L\subset Q\) is a linear subspace of dimension \(n-2\). The authors computes the intersection Poincaré polynomial of \(\mathbf{M}_n\) using Kirwan's desingularization method and the relation between \(\mathbf{M}_n\), the GIT quotient of the Kroneker quiver (Kontsevich's map space \(\mathbf{K}_n\)). Then they compute the intersection Poincaré polynomial of the moduli space of pure one-dimensional sheaves on the smooth surfaces \(\mathbb {P}^2\), \(\mathbb{F}_0\) and \(\mathbb {F}_1\).
Reviewer: Edoardo Ballico (Povo)On twists of smooth plane curves.https://www.zbmath.org/1456.111172021-04-16T16:22:00+00:00"Badr, Eslam"https://www.zbmath.org/authors/?q=ai:badr.eslam-e"Bars, Francesc"https://www.zbmath.org/authors/?q=ai:bars.francesc"Lorenzo García, Elisa"https://www.zbmath.org/authors/?q=ai:lorenzo-garcia.elisaLet \(C\) be a projective, smooth, non-hyperelliptic, curve and genus \(g \geq 3\) defined over a field \(k\). Denote by \(\bar{k }\) a fixed separable closure of \(k\) and by \(\bar{C}\) the curve \(C \times_k \bar{k}\). A twist of \(C\) over \(k\) is a projective, non-singular \(C^{\prime}\) defined over \(k\) with a \(\bar{k}\)-isomorphism \(\varphi : \overline{C^{\prime}} \rightarrow \bar{C}\). The paper under review deals with the following question: Assuming that \(C\) admits a smooth \(\bar{k}\)-plane model, does it have a smooth plane model over \(k\)? And if the answer is yes, does every twist of C over
\(k\) also have smooth plane model over \(k\)? The answer, in general, is negative. The twists possessing such models are characterized and an example of a twist not admitting any non-singular plane model over \(k\) is given. An interesting consequence is that explicit equations for a non-trivial Brauer-Severi surface are obtained. Furthermore, for smooth plane curves defined over \(k\) with a cyclic automorphism group generated by a diagonal matrix, a general theoretical result to
compute all its twists is presented.
Reviewer: Dimitros Poulakis (Thessaloniki)Bounds for the number of points on curves over finite fields.https://www.zbmath.org/1456.111132021-04-16T16:22:00+00:00"Arakelian, Nazar"https://www.zbmath.org/authors/?q=ai:arakelian.nazar"Borges, Herivelto"https://www.zbmath.org/authors/?q=ai:borges.heriveltoSummary: Let \(\mathcal{X}\) be a projective irreducible nonsingular algebraic curve defined over a finite field \(\mathbb{F}_q\). This paper presents a variation of the Stöhr-Voloch theory and sets new bounds to the number of \(\mathbb{F}_{q^r}\)-rational points on \(\mathcal{X}\). In certain cases, where comparison is possible, the results are shown to improve other bounds such as Weil's, Stöhr-Voloch's and Ihara's.On spectral curves and complexified boundaries of the phase-lock areas in a model of Josephson junction.https://www.zbmath.org/1456.340832021-04-16T16:22:00+00:00"Glutsyuk, A. A."https://www.zbmath.org/authors/?q=ai:glutsyuk.alexey-a"Netay, I. V."https://www.zbmath.org/authors/?q=ai:netay.igor-vA three-parameters family of special double confluent Heun equations, with real parameters \(l\), \(\lambda\), \(\mu\), is considered. The study of the real part of the spectral curve leads to applications to model of Josephson junction which is a family of dynamical systems on 2-torus depending on parameters \((B, A, \omega)\), where \(\omega\) is called the frequency. The authors provide an approach to study the boundaries of the phase-lock areas in \(\mathbb R^2(B, A)\) and their solutions, as \(\omega\) decreases to 0.They prove the irreducibility of the complex spectral curve \(\Gamma_l\) for every \(l\in\mathbb N\). They calculate its genus for \(l\le 20\) and present a conjecture on general genus formula. They apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. They show as well that its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and describe them. They prove that the spectral curve has no real ovals. Finally, they present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as \(\omega\) decreases, and a partial positive result towards its confirmation.
Reviewer: Bertin Zinsou (Johannesburg)Large automorphism groups of ordinary curves of even genus in odd characteristic.https://www.zbmath.org/1456.140372021-04-16T16:22:00+00:00"Montanucci, Maria"https://www.zbmath.org/authors/?q=ai:montanucci.maria"Speziali, Pietro"https://www.zbmath.org/authors/?q=ai:speziali.pietroSummary: Let \(\mathcal{X}\) be a (projective, nonsingular, geometrically irreducible) curve of even genus \(g(\mathcal{X})\ge 2\) defined over an algebraically closed field \(K\) of odd characteristic \(p\). If the \(p\)-rank \(\gamma(\mathcal{X})\) equals \(g(\mathcal{X})\) then \(\mathcal{X}\) is \textit{ordinary}. In this paper, we deal with \textit{large} automorphism groups \(G\) of ordinary curves of even genus. We prove that \(|G| < 821.37g(\mathcal{X})^{7/4}\) The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic by \textit{M. Giulietti} and \textit{G. Korchmáros} [Adv. Math. 349, 162--211 (2019; Zbl 1419.14040)]. According to this classification, for the exceptional cases \(\Aut(\mathcal{X}) \cong \mathrm{Alt}_7\) and \(\Aut \cong \mathrm{M}_{11}\) we show that the classical Hurwitz bound \(|\Aut(\mathcal{X})| \le 84(g(\mathcal{X})-1)\) holds, unless \(p = 3\), \(g(\mathcal{X})=26\) and \(\Aut(\mathcal{X}) \cong M_{11}\) an example for the latter case being given by the modular curve \(X(11)\) in characteristic 3.Categorical mirror symmetry on cohomology for a complex genus 2 curve.https://www.zbmath.org/1456.530702021-04-16T16:22:00+00:00"Cannizzo, Catherine"https://www.zbmath.org/authors/?q=ai:cannizzo.catherineSummary: Motivated by observations in physics, mirror symmetry is the concept that certain manifolds come in pairs \(X\) and \(Y\) such that the complex geometry on \(X\) mirrors the symplectic geometry on \(Y\). It allows one to deduce symplectic information about \(Y\) from known complex properties of \(X\). \textit{A. Strominger} et al. [Nucl. Phys., B 479, No. 1--2, 243--259 (1996; Zbl 0896.14024)] described how such pairs arise geometrically as torus fibrations with the same base and related fibers, known as SYZ mirror symmetry. \textit{M. Kontsevich} [in: Proceedings of the international congress of mathematicians, ICM '94, August 3-11, 1994, Zürich, Switzerland. Vol. I. Basel: Birkhäuser. 120--139 (1995; Zbl 0846.53021)] conjectured that a complex invariant on \(X\) (the bounded derived category of coherent sheaves) should be equivalent to a symplectic invariant of \(Y\) (the Fukaya category, see [\textit{D. Auroux}, Bolyai Soc. Math. Stud. 26, 85--136 (2014; Zbl 1325.53001); \textit{K. Fukaya} et al., Lagrangian intersection Floer theory. Anomaly and obstruction. I. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002); \textit{D. McDuff} et al., Virtual fundamental cycles in symplectic topology. New York, NY: American Mathematical Society (2019)]). This is known as homological mirror symmetry. In this project, we first use the construction of ``generalized SYZ mirrors'' for hypersurfaces in toric varieties following \textit{M. Abouzaid} et al. [Publ. Math., Inst. Hautes Étud. Sci. 123, 199--282 (2016; Zbl 1368.14056)], in order to obtain \(X\) and \(Y\) as manifolds. The complex manifold is the genus 2 curve \(\Sigma_2\) (so of general type \(c_1 < 0\)) as a hypersurface in its Jacobian torus. Its generalized SYZ mirror is a Landau-Ginzburg model \((Y, v_0)\) equipped with a holomorphic function \(v_0 : Y \to \mathbb{C}\) which we put the structure of a symplectic fibration on. We then describe an embedding of a full subcategory of \(D^b Coh(\Sigma_2)\) into a cohomological Fukaya-Seidel category of \(Y\) as a symplectic fibration. While our fibration is one of the first nonexact, non-Lefschetz fibrations to be equipped with a Fukaya category, the main geometric idea in defining it is the same as in Seidel's construction for Fukaya categories of Lefschetz fibrations in [\textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich: European Mathematical Society (EMS) (2008; Zbl 1159.53001); \textit{M. Abouzaid} and \textit{P. Seidel}, ``Lefschetz fibration methods in wrapped Floer cohomology'', in preparation].Genus 2 curves and generalized theta divisors.https://www.zbmath.org/1456.140392021-04-16T16:22:00+00:00"Brivio, Sonia"https://www.zbmath.org/authors/?q=ai:brivio.sonia"Favale, Filippo F."https://www.zbmath.org/authors/?q=ai:favale.filippo-francescoLet \({\mathcal U}_C(r,n)\) be the compactification of the moduli space of rank \(r\), degree \(n\) stable vector bundles on a projective complex curve \(C\), irreducible, smooth, of genus \(g \ge 2\). In the special case when \(n=r(g-1)\) it has a theta divisor \(\Theta _r\), defined as the ``natural Brill-Noether locus''.
For a fixed \(L \in \text{Pic}^{r(g-1)}(C)\) one has a moduli space of semistable vector bundles \({\mathcal S\mathcal U}_C(r,L)\) and a theta divisor \({\Theta }_{r,L}\).
Denote by \({\mathcal U}_C(r,n)\) the (compactification of) the moduli space (introduced by Seshadri) of rank \(r\), degree \(n\) stable vector bundles on \(C\). The main results of this paper are Theorems 2.5 and 3.4, for curves \(C\) of genus \(g\):
``There exists a vector bundle \(\mathcal V\) on \({\mathcal U}_C(r-1,r)\) of rank \(2r-1\) whose fibers at the point \([F]\in {\mathcal U}_C(r-1,r)\) is \(\mathrm{Ext}^1(F,{\mathcal O}_C)\). Let \({\mathbb P}({\mathcal V})\) be the associated projective bundle and \(\pi :{\mathbb P}({\mathcal V}) \rightarrow {\mathcal U}_C (r-1,r)\) the natural projection. Then the map \(\Phi : {\mathbb P}({\mathcal V}) \rightarrow {\Theta}_r\), sending \([v]\) to the vector bundle which is the extension of \(\pi ({v})\) by \({\mathcal O}_C\) , is a birational morphism''
and
``For a general stable bundle \(F \in {\mathcal S\mathcal U}_C(r,L)\) the map \[ \theta \circ \Phi \mid_{{\mathbb P}_F} : {\mathbb P}_F \rightarrow \mid r\Theta _M \mid \] is a linear embedding.''
In the second theorem \(\theta \) is the well known theta map and \({\mathbb P}_F=\pi ^{-1}([F])\).
The paper, which is very well written, explains the background, gives historical information and contains also other results which are of real interest.
Reviewer: Nicolae Manolache (Bucureşti)Hyperelliptic integrals modulo \(p\) and Cartier-Manin matrices.https://www.zbmath.org/1456.140362021-04-16T16:22:00+00:00"Varchenko, Alexander"https://www.zbmath.org/authors/?q=ai:varchenko.alexander-nSummary: The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field \(\mathbb{F}_p\) with a prime number \(p\) of elements were constructed only recently. In this paper we consider an example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus \(g\). It is known that in this case the total \(2g\)-dimensional space of holomorphic (multivalued) solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field \(\mathbb{F}_p\) in this case gives only a \(g\)-dimensional space of solutions, that is, a ``half'' of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field \(\mathbb{F}_p\) can be obtained by reduction modulo \(p\) of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier-Manin matrix of the hyperelliptic curve. That situation is analogous to an example of the elliptic integral considered in the classical paper [\textit{Yu. I. Manin}, Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802)].Hyperelliptic curves with maximal Galois action on the torsion points of their Jacobians.https://www.zbmath.org/1456.111182021-04-16T16:22:00+00:00"Landesman, Aaron"https://www.zbmath.org/authors/?q=ai:landesman.aaron"Swaminathan, Ashvin"https://www.zbmath.org/authors/?q=ai:swaminathan.ashvin-anand"Tao, James"https://www.zbmath.org/authors/?q=ai:tao.james"Xu, Yujie"https://www.zbmath.org/authors/?q=ai:xu.yujieSummary: In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-\(1\) subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus \(2\) and \(3\) over \(\mathbb{Q}\) whose Jacobians have such maximal adelic Galois representations.Reduced invariants from cuspidal maps.https://www.zbmath.org/1456.140702021-04-16T16:22:00+00:00"Battistella, Luca"https://www.zbmath.org/authors/?q=ai:battistella.luca"Carocci, Francesca"https://www.zbmath.org/authors/?q=ai:carocci.francesca"Manolache, Cristina"https://www.zbmath.org/authors/?q=ai:manolache.cristinaSummary: We consider genus one enumerative invariants arising from the Smyth-Viscardi moduli space of stable maps from curves with nodes and cusps. We prove that these invariants are equal to the reduced genus one invariants of the quintic threefold, providing a modular interpretation of the latter.Hypersurfaces with linear type singular loci.https://www.zbmath.org/1456.130082021-04-16T16:22:00+00:00"Farrahy, Amir Behzad"https://www.zbmath.org/authors/?q=ai:farrahy.amir-behzad"Nasrollah Nejad, Abbas"https://www.zbmath.org/authors/?q=ai:nasrollah-nejad.abbasThe authors consider an affine hypersurface \(X\subset\mathbb A^n\) defined by a polynomial \(f\) in the ring \(R=k[x_1,\dots,x_n]\) over an algebraically closed field \(k\) of characteristic \(0\). The singular locus of \(X\) is defined by the vanishing of the Jacobian ideal \(I(f)\), which is generated by \(f\) and the \(n\) derivatives of \(f\). The ideal \(J(f)\) generated by the derivatives alone is the gradient ideal of \(f\). The authors compare the schemes defined by \(I(f)\) and by \(J(f)\) around a singular point \(p\) of \(X\), assuming that \(X\) has only isolated singularities (so that, in particular, it is reduced). The Milnor number of \(X\) at \(p\) is the dimension of the localization \((R/J(f))_p\), while the Tjurina number of \(X\) at \(p\) is defined as the dimension of the localization \((R/I(f))_p\). \(X\) is called `locally Eulerian' if the Milnor number is equal to the Tjurina number at each singular point. The authors prove that \(X\) is locally Eulerian if and only if the symmetric and the Rees algebras of \(I(f)\) in \(R\) are isomorphic. In turn, the condition is equivalent to say that the Tjurina algebra \(R/I(f)\) is artinian Gorenstein. Then, the authors extend their study to the case of reduced projective plane curves \(C\), defining \(C\) to be `of gradient type' if the restriction of \(C\) to each affine chart associated to the singular points is locally Eulerian. The authors prove that curves \(C\) with only simple singularities are of gradient type. Furthermore, they prove that (reduced) quartics are of gradient type, while there are examples of quintics and sextics which are not of gradient type.
Reviewer: Luca Chiantini (Siena)Brill-Noether theory for curves of a fixed gonality.https://www.zbmath.org/1456.140382021-04-16T16:22:00+00:00"Jensen, David"https://www.zbmath.org/authors/?q=ai:jensen.david-w|jensen.david-l|jensen.david-d"Ranganathan, Dhruv"https://www.zbmath.org/authors/?q=ai:ranganathan.dhruvAuthors' abstract: ``We prove a generalisation of the Brill-Noether theorem for the variety of special divisors \(W^r_d(C)\) on a general curve \(C\) of prescribed gonality. Our main theorem gives a closed formula for the dimension of \(W^r_d(C)\). We build on previous work of \textit{N. Pflueger} [Adv. Math. 312, 46--63 (2017; Zbl 1366.14031)], who used an analysis of the tropical divisor theory of special chains of cycles to give upper bounds on the dimensions of Brill-Noether varieties on such curves. We prove his conjecture, that this upper bound is achieved for a general curve. Our methods introduce logarithmic stable maps as a systematic tool in Brill-Noether theory. A precise relation between the divisor theory on chains of cycles and the corresponding tropical maps theory is exploited to prove new regeneration theorems for linear series with negative Brill-Noether number. The strategy involves blending an analysis of obstruction theories for logarithmic stable maps with the geometry of Berkovich curves. To show the utility of these methods, we provide a short new derivation of lifting for special divisors on a chain of cycles with generic edge lengths, proved using different techniques by Cartwright, Jensen, and Payne [\textit{D. Cartwright} et al., Can. Math. Bull. 58, No. 2, 250--262 (2015; Zbl 1327.14266)]. A crucial technical result is a new realisability theorem for tropical stable maps in obstructed geometries, generalising a well-known theorem of \textit{D. E. Speyer} [Algebra Number Theory 8, No. 4, 963--998 (2014; Zbl 1301.14035)] on genus \(1\) curves to arbitrary genus.''
More precisely, the main result is the following:
Theorem. Let \(C\) be a general curve of genus \(g\) and gonality \(k\ge 2\) over the complex numbers. Assume that the quantity \(g-d+r\) is positive. Then \[\dim W^r_d(C)=\max_{\ell\in\{0,\dots,r'\}} g-(r-\ell+1)(g-d+r-\ell)-\ell k\] where \(r'=\min\{r, g-d + r-1\}\).
Reviewer: Edoardo Ballico (Povo)Components of the Hilbert scheme of smooth projective curves using ruled surfaces.https://www.zbmath.org/1456.140102021-04-16T16:22:00+00:00"Choi, Youngook"https://www.zbmath.org/authors/?q=ai:choi.youngook"Iliev, Hristo"https://www.zbmath.org/authors/?q=ai:iliev.hristo"Kim, Seonja"https://www.zbmath.org/authors/?q=ai:kim.seonjaLet \(H_{d,g} (\mathbb P^r)\) denote the Hilbert scheme parametrizing curves \(C \subset \mathbb P^r\) of degree \(d\) and arithmetic genus \(g\) and let \({\mathcal I}_{d,g,r} \subset H_{d,g} (\mathbb P^r)\) be the union of irreducible components whose general member is a smooth, irreducible, non-degenerate curve. \textit{L. Ein} showed that \({\mathcal I}_{d,g,r}\) is irreducible if \(d \geq g+r\) when \(r = 3\) [Ann. Sci. Éc. Norm. Supér. 19, 469--478 (1986; Zbl 0606.14003)] and \(r=4\) [\textit{L. Ein}, Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are various examples showing reducibility of \({\mathcal I}_{d,g,r}\) when \(d \geq g+r\) and \(r > 4\), disproving a claim of Severi.
Most of these examples were constructed with families of curves that are \(m\)-fold covers of \(\mathbb P^1\) with \(m \geq 3\), but the authors gave an example with a family of curves that are double covers of irrational curves [Taiwanese J. Math. 21, 583--600 (2017; Zbl 1390.14019)].
Here the authors reconstruct their example in a more geometric way as a family \(\mathcal D\) of curves on ruled surfaces. The new construction allows them to show that \(\mathcal D\) is generically smooth of expected dimension, hence a regular component. When including the distinguished component dominating \(\mathcal M_g\), this gives the first examples of Hilbert schemes \({\mathcal I}_{d,g,r}\) satisfying \(d \geq g+r\) with \textit{two} regular components.
Reviewer: Scott Nollet (Fort Worth)On generalized parabolic Hitchin pairs.https://www.zbmath.org/1456.140402021-04-16T16:22:00+00:00"Das, Sourav"https://www.zbmath.org/authors/?q=ai:das.souravLet \(Y\) be a nodal curve with a single node \(p\) and normalisation a smooth curve \(X\). Let \(p_1, p_2\) be points of \(X\) lying over \(p\). Let \(L\) be a line bundle on \(Y\) and \(L_0\) its pull back to \(X\). A generalised parabolic (\(L_0\)-twisted) Hitchin pair (GPH in short) of rank \(n\) and degree \(d\) is a Hitchin pair \((E,\phi)\) together with a generalised parabolic structure. We recall that \(E\) is a vector bundle of rank \(n\) and degree \(d\) and \(\phi: E \to E \otimes L_0\) is a homomorphism. A generalised parabolic structure consists of a subspace \(F(E) \subset E_{p_1} \oplus E_{p_2}\) of dimension \(n\) with a weight \(\alpha, 0 < \alpha \le 1\). Bhosle had constructed the moduli space \(M_{GPH}\) of \(\alpha\)-semistable GPH and given a proper Hitchin map from it to an affine space \(A\). A GPH is called a good GPH if it preserves \(\phi\), let \(M^{\alpha-st}_{GPH}\) be the moduli space of good \(\alpha\)-stable GPH. There is a morphism \(q: M^{\alpha-st}_{GPH} \to M^{st}_{TFH},\) the latter being the moduli space of stable torsion free (\(L\)-twisted) Hitchin pairs on \(Y\).
The author defines a (stable) Gieseker Hitchin pair data (GHPD in short) generalising the notion of (stable) Gieseker vector bundles by Balaji et al. In case the rank \(n\) and degree \(d\) are coprime, he constructs the moduli space \(M^{st}_{GHPD}\) of these stable objects and shows that it is a normalisation of the moduli space \(M^{\alpha-st}_{GHP}\) of stable Gieseker Hitchin pairs. He shows that the Hitchin map on \(M^{st}_{GHPD}\) is proper. There is a natural morphism \(\pi_0: M^{st}_{GHP} \to M_{TFH}\). The author gives morphisms of moduli schemes \(\pi: M^{st}_{GHPD} \to M^{\alpha-st}_{GHP}\) and \(q_0: M^{st}_{GHPD} \to M^{\alpha-st}_{GPH}\) (for \(\alpha\) close to \(1\) or equal to \(1\)) such that \(\pi_0 \circ \pi = q \circ q_0\).
Reviewer: Usha N. Bhosle (Bangalore)Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities.https://www.zbmath.org/1456.580242021-04-16T16:22:00+00:00"Kalvin, Victor"https://www.zbmath.org/authors/?q=ai:kalvin.victorSummary: We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area \(4 \pi\) the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomon formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.Counting Richelot isogenies between superspecial abelian surfaces.https://www.zbmath.org/1456.140552021-04-16T16:22:00+00:00"Katsura, Toshiyuki"https://www.zbmath.org/authors/?q=ai:katsura.toshiyuki"Takashima, Katsuyuki"https://www.zbmath.org/authors/?q=ai:takashima.katsuyukiSummary: \textit{W. Castryck} et al. [``Hash functions from superspecial genus-2 curves using Richelot isogenies'', J. Math. Crypt. 14, 268--292 (2020; \url{doi:10.1515/jmc-2019-0021})] used superspecial genus-2 curves and their Richelot isogeny graph for basing genus-2 isogeny cryptography, and recently, \textit{C. Costello} and \textit{B. Smith} [``The supersingular isogeny problem in genus 2 and beyond'', in: Post-quantum cryptography. Cham: Springer. 151--168 (2020)], devised an improved isogeny path-finding algorithm in the genus-2 setting. In order to establish a firm ground for the cryptographic construction and analysis, we give a new characterization of decomposed Richelot isogenies in terms of involutive reduced automorphisms of genus-2 curves over a finite field, and explicitly count such decomposed (and nondecomposed) Richelot isogenies between superspecial principally polarized abelian surfaces. As a corollary, we give another algebraic geometric proof of Theorem 2 in the paper of Castryck et al. [loc. cit.].
For the entire collection see [Zbl 1452.11005].Enumerative geometry of plane curves.https://www.zbmath.org/1456.140692021-04-16T16:22:00+00:00"Caporaso, Lucia"https://www.zbmath.org/authors/?q=ai:caporaso.luciaThe author gives a nice account of some results in the enumerative geometry of plane curves. After explaining how curves \(C \subset \mathbb P^2\) of degree \(d\) are parametrized by projective space \(P_d = \mathbb P^{c_d}\) with \(c_d = d(d+3)/2\) via their coefficients,
she gives a very readable proof of the fact that the locus \(S_d \subset P_d\) corresponding to singular curves is a hypersurface of degree \(3(d-1)^2\) in the second section. Then she moves on to consider the more refined \textit{Severi variety} \(S_{d,\delta} \subset P_d\) parametrizing degree \(d\) curves with at least \(\delta\) nodes, explaining why \(S_{d, \delta}\) is empty for \(\delta > \binom{d-1}{2}\). For \(\delta \leq \binom{d-1}{2}\), \textit{J. Harris} [Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)] showed that \(S_{d, \delta}\) is irreducible of dimension \(c_d - \delta\) and the general curve in this family has geometric genus \(g=\binom{d-1}{2} - \delta\). Next the author considers the general problem of computing the number of irreducible plane curves of degree \(d\) and genus \(g\) passing through \(3d+g-1\) general points, explaining the complicated recursive formula of \textit{M. Kontsevich} in the case \(g=0\) [Prog. Math. 129, 335--368 (1995; Zbl 0885.14028)], which originally came from physical considerations as a condition characterizing the associativity of the quantum product on quantum cohomology. For curves of genus \(g>0\), she gives a few examples, but the full story is explained in work of \textit{L. Caporaso} and \textit{J. Harris} [Invent. Math. 131, No. 2, 345--392 (1998; Zbl 0934.14040)].
Reviewer: Scott Nollet (Fort Worth)On the bielliptic and bihyperelliptic loci.https://www.zbmath.org/1456.140302021-04-16T16:22:00+00:00"Frediani, Paola"https://www.zbmath.org/authors/?q=ai:frediani.paola"Porru, Paola"https://www.zbmath.org/authors/?q=ai:porru.paolaLet \(\mathcal {A}_g\) be the moduli space of \(g\)-dimensional principallly polarized Abelian varieties and \(j: \mathcal {M}_g\to \mathcal {A}_g\) the Torelli map. \(\mathcal{A}_g\) has a metric, the Siegel metric, and a conjecture of Coleman and Oort is related to totally geodesic subvarieties for the Siegel metric. \(\mathcal {M}_g\) get the metric and the second Gaussian map may be used to detect its totally geodesic subvarieties [\textit{F. Colombo} and \textit{P. Frediani}, Mich. Math. J. 58, No. 3, 745--758 (2009; Zbl 1191.14030); Trans. Amer. Math. Soc. 362, No. 3, 745--758 (2010; Zbl. 1191.14030); \textit{E. Colombo} et al., Int. J. Math. 26, No. 1, Article ID 15500005 (2015; Zbl. 1312.14076)]). In this paper the authors prove that for \(g\ge 4\) the locus of bielliptic curves is not totally geodesic in \(\mathcal {A}_f\), contrary to the case \(g=3\), and that it is not totally geodesic the locus \(\mathcal {BH}_{g,g'}\) of double coverings of genus \(g\) of hyperelliptic curves of genus \(g'\) (double hyperelliptic curves) for \(g \ge 3g'\ge 6\).
Reviewer: Edoardo Ballico (Povo)An application of the Hasse-Weil bound to rational functions over finite fields.https://www.zbmath.org/1456.112282021-04-16T16:22:00+00:00"Hou, Xiang-Dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dong"Iezzi, Annamaria"https://www.zbmath.org/authors/?q=ai:iezzi.annamariaLet \(\mathbb{F}_q\) be the finite field of \(q\) elements. In the paper under review, an application of the Aubry-Perret bound (a generalization of the Hasse-Weil bound for smooth curves) for an algebraic projective, absolutely irreducible and non-singular curve defined over \(\mathbb{F}_q\) is studied [\textit{Y. Aubry} and \textit{M. Perret}, in: Arithmetic, geometry, and coding theory. Proceedings of the international conference held at CIRM, Luminy, France, 1993. Berlin: Walter de Gruyter, 1--7 (1996; Zbl 0873.11037)]. More precisely, the authors prove that if \(f, g \in\mathbb{F}_q(X)\) are non-constant rational functions with degree \(d\) and \(\delta\) respectively such that \(q \geq (d+ \delta)^4\), \(f(\mathbb{F}_q) \subset g (\mathbb{F}_q \cup \{ \infty\})\) and for all \(a \in \mathbb{F}_q \cup \{ \infty\}\) with at most \(8(d+ \delta)\) exceptions, \(|\{x \in \mathbb{F}_q: g(x)=g(a)\}| > \frac{\delta}{2}\), then there exists a rational function \(h \in \mathbb{F}_q(X)\) such that \(f=g \circ h.\)
This application of the Aubry-Perret bound to rational functions on finite fields is motivated by two special cases related to the study of permutation polynomials: \(g(X)=X^2+X\) when the characteristic of \(\mathbb{F}_q\) is \(2\) and \(g(X)=X^2\) when the characteristic of \(\mathbb{F}_q\) is \(3\) [\textit{X.-D. Hou}, Cryptogr. Commun. 11, No. 6, 1199--1210 (2019; Zbl 1446.11206)] and [\textit{X.-D. Hou} et al., Finite Fields Appl. 61, Article ID 101596, 27 p. (2020; Zbl 07160767)].
Finally, using an explicit estimate for absolutely irreducible multivariate polynomials with coefficients in \(\mathbb{F}_q\) [\textit{A. Cafure} and \textit{G. Matera}, Finite Fields Appl. 12, No. 2, 155--185 (2006; Zbl 1163.11329)], a generalization to multivariate rational functions is also included.
Reviewer: Mariana Pérez (Buenos Aires)Arithmetic topology in Ihara theory. II: Milnor invariants, dilogarithmic Heisenberg coverings and triple power residue symbols.https://www.zbmath.org/1456.112162021-04-16T16:22:00+00:00"Hirano, Hikaru"https://www.zbmath.org/authors/?q=ai:hirano.hikaru"Morishita, Masanori"https://www.zbmath.org/authors/?q=ai:morishita.masanoriIn this paper, a wide class of triple quadratic (resp., cubic) residue symbols \([p_1,p_2,p_3]\) of primes \(p_i\) (\(i=1,2,3\)
in \(\mathbb{Q}\) (resp., \(\mathbb{Q}(\sqrt{-3})\)) is connected to the mod \(\ell\) Milnor invariants introduced in previous work [\textit{H. Kodani} et al., Publ. Res. Inst. Math. Sci. 53, No. 4, 629--688 (2017; Zbl 1430.11082)] as certain coefficients of Magnus series of Frobenius elements arising from Ihara theory on Galois representations in the pro-\(\ell\) fundamental groups of punctured projective lines. Dilogarithmic mod \(\ell\) Heisenberg ramified covering \(D(\ell)\) of \(\mathbb{P}^1\) plays a central role ``as a higher analog of the dilogarithmic function for the gerbe associated to the mod \(\ell\) Heisenberg group''.
The monodromy transformations of certain functions on \(D(\ell)\) along the pro-\(\ell\) longitudes of Frobenius elements
turn out to capture the aimed power residue symbols via Wojtkowiak's work on the \(\ell\)-adic Galois polylogarithms.
Reviewer: Hiroaki Nakamura (Osaka)