Warwick Algebraic Geometry Seminar

Abstracts

Christian Böhning (Warwick) - Stable nonrationality of ℚ-Fano threefolds with anticanonical models of small codimension

We report on joint work with Hans-Christian v. Bothmer (still very much in progress) on rationality properties of some classes of ℚ-Fano threefolds given in a Gorenstein format in the sense of Reid et al. Besides proving stable non-rationality for very general members of families of such Fanos in new cases, this provides a unified view on and recovers results of Ahmadinezhad-Okada-Totaro et al. Time permitting, we will in addition say a few words about singular models of cubics in the context of the degeneration method.

Fabien Cléry (Loughborough) - Vector-valued Siegel modular forms of degree 2 with character

The link between covariants of binary sextics and vector-valued Siegel modular forms of degree 2, obtained in a previous work with G. van der Geer and C. Faber, is an efficient way for producing such modular forms. By using this link, the structure of some graded rings of vector-valued Siegel modular forms of degree 2 with character will be given. This talk is based on a joint work with G. van der Geer and C. Faber.

Michel van Garrel (Warwick) - On a conjecture by Voskresenskii

In this collaboration with M. Florence, we are interested in settings where the notions of rationality and stable rationality become equivalent. We explore this question for tori over any field, where the equivalence of the two notions was conjectured by Voskresenskii. For a certain class of tori, this conjecture was proven by Klyachko by means of general principles. We give a new explicit proof of this result by exhibiting simple morphisms. This proof has an application in cryptography.

Alessio D'Alì (Warwick) - Generalized Stanley-Reisner rings and translative group actions

The Stanley-Reisner correspondence, which assigns a commutative ring to each finite simplicial complex, is a useful and well-studied bridge between commutative algebra and combinatorics. In 1987 Sergey Yuzvinsky proposed a construction that allows to see the Stanley-Reisner ring of a finite simplicial complex as the ring of global sections of a sheaf of rings on a poset. Motivated by applications in the theory of Abelian arrangements, Emanuele Delucchi and I extend Yuzvinsky's construction to the case of (possibly infinite) finite-length simplicial posets. This generalization behaves well with respect to quotients of simplicial complexes and posets by translative group actions.

Mattia Talpo (Imperial) - Topological realization of varieties over ℂ((t)) via log geometry

I will describe some work in progress with Piotr Achinger about defining a “Betti realization” functor for varieties over the formal punctured disk Spec ℂ((t)), i.e. defined by polynomials with coefficients in the field of formal Laurent series in one variable. Our construction is via “good models” over the power series ring ℂ[[t]] and the “Kato-Nakayama” construction in logarithmic geometry, that I will review during the talk.

Ehud Meir (Aberdeen) - Applications of GIT to Hopf algebras

In this talk I will describe applications of Geometric Invariant Theory (GIT) to the study of finite dimensional Hopf algebras and related algebraic structures. If time permits, I will also explain a related classification result in the theory of finite group schemes and a non-commutative geometry generalisation of them, known as Hopf orders.

Jeongseok Oh (KIAS) - Localized Chern characters for 2-periodic complexes

For a two-periodic complex of vector bundles, Polishchuk and Vaintrob have constructed its localized Chern character. We explore some basic properties of this localized Chern character. In particular, we show that the cosection localization defined by Kiem and Li is equivalent to a localized Chern character operation for the associated two-periodic Koszul complex, strengthening a work of Chang, Li, and Li. We apply this equivalence to the comparison of virtual classes of moduli of ε-stable quasimaps and moduli of the corresponding LG ε-stable quasimaps. This is based on joint work with Bumsig Kim. Furthermore, if the time is permitted, we discuss the comparison of virtual structure sheaves either. The latter is joint work with B. Sreedhar.

Stefan Schreieder (LMU München) - Stably irrational hypersurfaces of small slopes

We explain the rationality problem for hypersurfaces in projective space and review what is known about this problem. We then explain how to prove that over any uncountable field of characteristic different from two, a very general hypersurface of dimension n > 2 and degree at least 2+log₂(n) is not stably rational. This significantly improves earlier results of Kollár and Totaro.

René Mboro (University of Bonn) - Remarks on varieties of essential CH₀-dimension at most 2

We present the notion, due to Voisin, of essential CH₀-dimension of a variety and give two flavors of sufficient conditions for a variety with trivial group of 0-cycles and essential CH₀-dimension at most 2 to have, in fact, essential CH₀-dimension 0 (i.e. to admit a decomposition of the diagonal).