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Algebraic Geometry Seminar 21/22 Term 2

The algebraic geometry seminar in Term 2 2021/22 will usually meet on Wednesdays at 3pm, though we will sometimes change to allow speakers from other time zones.

See the talks from last year here.

Wednesday 2 February 2022, 3pm. Speaker: Chunyi Li (Warwick)

Title: Bridgeland stability conditions on the Hilbert scheme of K3 surfaces.

Abstract: Let E be an elliptic curve and S be the Kummer K3 surface associated with E\times E. We construct a family of stability conditions on Hilb^nS with the central charges -ch_{2n}^{b+ia} parameterized by a>0 and b. In the talk, I will explain the background of the construction of stability conditions and some recent techniques introduced in the project. This is an ongoing project joint with Emanuele Macrì, Paolo Stellari, and Xiaolei Zhao.

Wednesday 9 February 2022, 3pm. Speaker: Kristian Ranestad (Oslo)

Title: Betti tables of quaternary quartics and arithmetic Gorenstein Calabi-Yau threefolds in P7.

Abstract: The possible betti tables of the apolar ideal of quartic forms in four variables define a stratification of the space of such forms. An Artinian reduction by linear forms of the ideal of an arithmetic Gorenstein Calabi-Yau threefold in P7 is the apolar ideal of a quartic form. A natural question is which quartic forms appear this way from possibly reducible CY threefolds. I will report on recent work with G. and M. Kapustka, M. Stilmann, H. Schenk and B. Yuan which recovers and extends some recent results by Patience Ablett.

Wednesday 16 February 2022, 3pm. Speaker: Bivas Khan (soon to be Warwick)

Title: On (semi)stability of equivariant vector bundle on toric varieties

Abstract: The study of equivariant vector bundles on toric varieties is one of those subjects which lie on the crossroads of Geometry, Algebra and Combinatorics. It turns out that natural vector bundles over toric varieties, such as the various tautological bundles, tangent and cotangent bundles and their tensor products, exterior products, symmetric products etc., are equivariant. In this talk, we review the combinatorial description of equivariant vector bundles initiated by Klyachko. We then give a combinatorial criterion of (semi)stability of equivariant torsion-free sheaves with respect to a given polarization. As an application, we obtain a complete answer to (semi)stability of tangent bundle of X with Picard number 2 and toric Fano 4-folds with Picard number ?^ɤ 3. This is joint work with Jyoti Dasgupta and Arijit Dey.

Wednesday 23 February, 3pm. Speaker: Vlerë Mehmeti (Paris-Saclay)

Title: Local global principles and non-Archimedean analytic curves

Abstract: I will be speaking of an application of non-Archimedean analytic geometry to questions related to the existence of rational points on varieties. More precisely, several local-global principles applicable to quadratic forms will be presented, all of them obtained by working over Berkovich analytic curves. The main tool I employ is an adaptation of the so called patching technique, which has lately become an important method for the study of such questions. The talk will begin with a brief introduction of the main notions.

Wednesday 2 March, Double-header.

3pm Speaker: Daniele Turchetti (Warwick)

Title: Models of curves over DVRs via Berkovich geometry

Abstract: The theory of models of varieties over discrete valuation rings is an important tool for topics such as deformation theory, moduli spaces, and degenerations. In the 1960s, Deligne and Mumford proved that any smooth projective curve C over a discretely valued field K has a semi-stable model after base-change to a finite Galois extension L|K. The question of determining such extension has been investigated ever since but has been settled only in the case where L|K is tamely ramified.

In this talk, I will present two results on the behaviour of models of curves under base change. The first (joint with Lorenzo Fantini) exploits the geometry of the Berkovich analytification of C to describe the extension L|K in terms of regular models; the second (joint with Andrew Obus) investigates more in detail the case of potentially multiplicative reduction yielding new results in the case where L|K is wildly ramified.

4:15 Speaker: George Shabat (Moscow State University, Independent University of Moscow, Russian State University for the Humanities)

Title: Dessins d'enfants and moduli spaces of curves

Abstract: A very brief overview of the dessins d'enfants theory will be given. Two kinds of its relations with the moduli spaces of curves will be outlined. (1) The Mumford-Penner-Kontsevich-... construction of cell decomposition of (decorated) moduli spaces and its arithmetic version (Mulase-Penkava). (2) The critical filtrations of Hurwitz and moduli spaces, introduced by the speaker, will be discussed, some examples given and open questions formulated.

Wednesday 9 March, 3pm. Speaker: Zheng Hua

Title: Some results on Feigin-Odesskii Poisson structures

Abstract: Feigin-Odesskii Poisson structures are a class of quadratic Poisson structures on projective spaces of arbitrary dimensions. They were constructed by Feigin and Odesskii in 80s as semi-classical limits of elliptic algebras. In this talk I will present a construction of Poisson structure on moduli space of complexes of vector bundles over Calabi-Yau curves. Feigin-Odesskii Poisson structures are realized as special cases. Using the techniques of moduli stack, we prove several new results on Feigin-Odesskii Poisson structures including classification of symplectic leaves, bihamiltonian structure etc.. This is based on a series of joint work with Alexander Polishchuk.

Wednesday 16 March, 12pm. Speaker: Leonid Monin (MPIM MiS) (note unusual time)

Title: Inversion of matrices, ML-degrees and the space of complete quadrics

Abstract: What is the degree of the variety L-1 obtained as the closure of the set of inverses of matrices from a generic linear subspace L of symmetric matrices of size nxn? Although this is an interesting geometric question in its own right, it is also motivated by algebraic statistics: the degree of L-1 is equal to the maximum likelihood degree (ML-degree) of a generic linear concentration model. In 2010, Sturmfels and Uhler computed the ML-degrees for dim(L) less than 5 and conjectured that for the fixed dimension of L the ML-degree is a polynomial in n. In my talk I will describe geometric methods to approach the computation of ML-degrees which in particular allow to prove the polynomiality conjecture. The talk is based on a joint work with Laurent Manivel, Mateusz Michalek, Tim Seynnaeve, Martin Vodicka, and Jaroslaw A. Wisniewski.