Recent zbMATH articles in MSC 14Ehttps://www.zbmath.org/atom/cc/14E2021-04-16T16:22:00+00:00WerkzeugThe group of symplectic birational maps of the plane and the dynamics of a family of 4D maps.https://www.zbmath.org/1456.530662021-04-16T16:22:00+00:00"Cruz, Inês"https://www.zbmath.org/authors/?q=ai:cruz.ines"Mena-Matos, Helena"https://www.zbmath.org/authors/?q=ai:mena-matos.helena"Sousa-Dias, Esmeralda"https://www.zbmath.org/authors/?q=ai:sousa-dias.esmeraldaSummary: We consider a family of birational maps \(\varphi_k\) in dimension 4, arising in the context of cluster algebras from a mutation-periodic quiver of period 2. We approach the dynamics of the family \(\varphi_k\) using Poisson geometry tools, namely the properties of the restrictions of the maps \(\varphi_k\) and their fourth iterate \(\varphi^{(4)}_k\) to the symplectic leaves of an appropriate Poisson manifold \((\mathbb{R}^4_+, P)\). These restricted maps are shown to belong to a group of symplectic birational maps of the plane which is isomorphic to the semidirect product \(SL(2, \mathbb{Z})\ltimes\mathbb{R}^2\). The study of these restricted maps leads to the conclusion that there are three different types of dynamical behaviour for \(\varphi_k\) characterized by the parameter values \(k = 1\), \(k = 2\) and \(k\geq 3\).Moderately ramified actions in positive characteristic.https://www.zbmath.org/1456.140062021-04-16T16:22:00+00:00"Lorenzini, Dino"https://www.zbmath.org/authors/?q=ai:lorenzini.dino-j"Schröer, Stefan"https://www.zbmath.org/authors/?q=ai:schroer.stefanSummary: In characteristic 2 and dimension 2, wild \(\mathbb{Z}/2\mathbb{Z} \)-actions on \(k[[u, v]]\) ramified precisely at the origin were classified by \textit{M. Artin} [Proc. Am. Math. Soc. 52, 60--64 (1975; Zbl 0315.14015)], who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic \(p>0\) and dimension \(n\ge 2\) arising from certain non-linear actions of \(\mathbb{Z}/p\mathbb{Z}\) on the formal power series ring \(k[[u_1,\dots,u_n]]\). These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when \(p=2\). In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.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)Finite groups of bimeromorphic selfmaps of uniruled Kähler threefolds.https://www.zbmath.org/1456.140202021-04-16T16:22:00+00:00"Prokhorov, Yuri G."https://www.zbmath.org/authors/?q=ai:prokhorov.yuri-g"Shramov, Constantin A."https://www.zbmath.org/authors/?q=ai:shramov.konstantin-aA group \(\Gamma\) is called Jordan if there exists a constant \(C\) such that if \(G\) is a finite subgroup of \(\Gamma\) then \(G\) has a normal abelian subgroup of index at most \(C\). The authors classified in their previous works compact complex surfaces and projective threefolds whose groups of bimeromorphic transformations are Jordan. In the paper under review they proved that the only non-algebraic compact Kähler threefolds whose groups of bimeromorphic transformations are not Jordan are projectivizations of holomorphic vector bundles of rank two over two-dimensional complex tori of algebraic dimension one. With the help of their previous results on surfaces and projective threefolds, the proof is reduced to the study of bimeromorphic self maps of conic bundles over non-algebraic surfaces.
Reviewer: Shengyuan Zhao (Stony Brook)Idealistic exponents: tangent cone, ridge, characteristic polyhedra.https://www.zbmath.org/1456.140072021-04-16T16:22:00+00:00"Schober, Bernd"https://www.zbmath.org/authors/?q=ai:schober.berndLet \(X\) be a regular irreducible scheme of finite type over a field \(k\).
Studying the resolution of singularities Hironaka introduced idealistic exponents \(\mathbb E=(J,b)\) on \(X\) as a pair consisting of a quasi-coherent ideal sheaf \(J\) on \(X\) and a positive integer \(b\in \mathbb Z\).
Based on Hironaka's work idealistic exponents are studied over Spec\((\mathbb Z)\). An idealistic interpretation of the tangent cone, the directrix and the ridge
is given. The notion of characteristic polyhedra od idealistic exponents is given and studied.
Reviewer: Gerhard Pfister (Kaiserslautern)Signature morphisms from the Cremona group over a non-closed field.https://www.zbmath.org/1456.140192021-04-16T16:22:00+00:00"Lamy, Stéphane"https://www.zbmath.org/authors/?q=ai:lamy.stephane"Zimmermann, Susanna"https://www.zbmath.org/authors/?q=ai:zimmermann.susannaThe Cremona group \(\operatorname{Cr}_2(\Bbbk)\) is the group of birational transformations of the projective plane \(\mathbb{P}^2\) over a field \(\Bbbk\). It was a long-standing question whether \(\operatorname{Cr}_n(\mathbb{C})\) is simple group. Several years ago \textit{S. Cantat} et al. made a breakthrough in [Acta Math. 210, No. 1, 31--94 (2013; Zbl 1278.14017)] by proving that \(\operatorname{Cr}_2(\mathbb{C})\) is not simple. \textit{A. Lonjou} generalized the result to an arbitrary field in [Ann. Inst. Fourier 66, No. 5, 2021--2046 (2016; Zbl 1365.14017)]. Over the field of complex numbers it was classically known that \(\operatorname{Cr}_2(\Bbbk)\) does not admit any non trivial homomorphism to an abelian group. Over the field of real numbers \textit{S. Zimmermann} proved in [Duke Math. J. 167, No. 2, 211--267 (2018; Zbl 1402.14015) ] that the abelianization of \(\operatorname{Cr}_2(\mathbb{R})\) is a direct sum of uncountably many \(\mathbb{Z}/2\mathbb{Z}\).
The article under review deals with an arbitrary perfect field with at least one Galois extension of degree eight. The authors constructed a tree on which \(\operatorname{Cr}_2(\Bbbk)\) acts so that \(\operatorname{Cr}_2(\Bbbk)\) can be written as an amalgam product by Bass-Serre theory. Note that each factor in the amalgam product is a big group and there are a lot of factors (same cardinality as the field \(\Bbbk\)). Consequently the authors constructed a homomorphism from \(\operatorname{Cr}_2(\Bbbk)\) to a free product of \(\mathbb{Z}/2\mathbb{Z}\), thus also a homomorphism from \(\operatorname{Cr}_2(\Bbbk)\) to a direct sum of \(\mathbb{Z}/2\mathbb{Z}\).
The tree mentionned above comes from a square complex constructed in this paper on which \(\operatorname{Cr}_2(\Bbbk)\) acts. The vertices of the square are rank \(r\) fibrations with \(r=1,2,3\); rank \(r\) fibrations are generalizations of Mori fiber spaces. Roughly speaking the edges and the faces of the square complex record Sarkisov links and relations among Sarkisov links. If we blow up a general point of degree eight on \(\mathbb P^2\) then we obtain a del Pezzo surface of degree \(1\). Such a del Pezzo surface gives a rank \(2\) fibration and an element in \(\operatorname{Cr}_2(\Bbbk)\) called a Bertini involution. This is where the hypothesis on the field \(\Bbbk\) is used. Rougly speaking the tree is constructed by recording the action of \(\operatorname{Cr}_2(\Bbbk)\) on the part of the square complex containing these Bertini involutions.
Reviewer: Shengyuan Zhao (Stony Brook)Birational superrigidity and \(K\)-stability of Fano complete intersections of index \(1\).https://www.zbmath.org/1456.140502021-04-16T16:22:00+00:00"Zhuang, Ziquan"https://www.zbmath.org/authors/?q=ai:zhuang.ziquanA Fano variety \(X\) is said to be birationally superrigid if it has terminal singularities,
it is \({\mathbb Q}\)-factorial of Picard number 1, and every birational map \(X\)
to a Mori fiber space is an isomorphism. On the other hand, \(X\) is \(K\)-stable
with respect to its anticanonical bundle if, essentially, it admits a Kähler-Einstein
metric, and \(K\)-stability is encoded in the positivity of the invariants \(\beta(F)\)
for \(F\) any dreamy prime divisor \(F\) over \(X\) (see 2.2 for details). In the paper under
review the author shows (see Thm. 1.2 and 1.3) that for a \(n\)-dimensional smooth Fano complete
intersection \(X \subset {\mathbb P}^{n+r}\) of index one, if \(n \geq 10r\) then \(X\) is birationally
superrigid and \(K\)-stable. Moreover, the smooth complete intersection of a quadric and a cubic
in \({\mathbb P}^5\) is also \(K\)-stable. For a Fano manifold (see Def. A.1 in the appendix of
the paper under review) \(X\) is said to be conditionally birationally superrigid if
every birational map from \(X\) to a Mori fiber space whose undefined locus has
codimension at least \(1\) plus the index of \(X\) is an isomorphism.
In the Appendix, the authors show that Fano complete intersections of higher index in large dimension (see Cor. A.3
for details) are conditionally birationally superrigid.
Reviewer: Roberto Muñoz (Madrid)Birational models of moduli spaces of coherent sheaves on the projective plane.https://www.zbmath.org/1456.140162021-04-16T16:22:00+00:00"Li, Chunyi"https://www.zbmath.org/authors/?q=ai:li.chunyi"Zhao, Xiaolei"https://www.zbmath.org/authors/?q=ai:zhao.xiaoleiThe thema of the moduli spaces of sheaves on surfaces is a field of intense interest and study. In this very consistent paper one deals with coherent sheaves on the complex projective plane, in the frame of Bridgeland stability. The paper consists of 4 sections: \newline In section 1 classical results about stable sheaves on the projective plane are reviewed, mainly results of Drezet and Le Potier, and preparatory lemmas for the next sections are proved. \newline In section 2 one proves that the moduli space \({\mathcal M}^s_\sigma (w)\), where \(\sigma \) is a stability condition and \(w\) is a character in the Grothendieck group \(K({\mathbb P}^2)\) of \(D^b({\mathbb P}^2)\), ``is smooth and irreducible for generic \(\sigma \) and primitive \(w\)''.
In Section 3 one computes the last wall, and then one obtains a criterion for actual walls of \({\mathcal M}^s_\sigma (w)\) \newline In Section 4 one uses the above criterion for computing ``the nef and movable cone boundaries'' and one presents the example of the Chern character \((4,0,-15)\). \newline One has to note that the paper gives many references to existing literature on the subject, together with a discussion of the correlations to the paper under review. In this way it can be considered also a short challanging guide.
Reviewer: Nicolae Manolache (Bucureşti)Rationality problem for norm one tori in small dimensions.https://www.zbmath.org/1456.110432021-04-16T16:22:00+00:00"Hasegawa, Sumito"https://www.zbmath.org/authors/?q=ai:hasegawa.sumito"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 \(n-1\) for \(n=2^e (e\geq 1)\) as a power of \(2\) and \(n=12, 14, 15\). Retract non-rationality of norm one tori for primitive \(G\leq S_{2p}\) where \(p\) is a prime number and for the five Mathieu groups \(M_n\leq S_n (n=11,12,22,23,24)\) is also given.