# Algebraic Geometry Seminar 22/23 Term 1

The algebraic geometry seminar in Term 1 2022/2023 will usually meet on Wednesdays at 3pm in B3.02, though we may sometimes change to allow speakers from other time zones.

See the talks from last term here.

**Wednesday 19 October 2022, 3pm. Speaker: Nivedita Viswanathan (Loughborough)**

Title: On K-stability of some singular del Pezzo surfaces

Abstract: There has been a lot of development recently in understanding the existence of Kahler-Einstein metrics on Fano manifolds due to the Yau-Tian-Donaldson conjecture, which gives us a way of looking at this problem in terms of the notion of K-stability. In particular, this problem is solved in totality for smooth del Pezzo surfaces by Tian. For del Pezzo surfaces with quotient singularities, there are partial results. In this talk, we will consider singular del Pezzo surfaces which are quasi-smooth, well-formed hypersurfaces in weighted projective space, and understand what we can say about their K-stability. This is ongoing joint work with In-Kyun Kim and Joonyeong Won.

**Wednesday 26 October 2022, 3pm. Speaker: Arman Sarikyan (Edinburgh)**

Title: On the Rationality of Fano-Enriques Threefolds

Abstract: A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample Cartier divisor that is an Enriques surface with at most canonical singularities. There is no complete classification of Fano-Enriques threefolds yet. However, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical coverings, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.

**Wednesday 2 November 2022, 3pm. Speaker: Michel van Garrel (Birmingham)**

Title: Log mirror symmetry

Abstract: Start with a smooth Fano variety X and a smooth anticanonical divisor D. Consider the problem of counting maps from the projective line to X that meet D along a curve in only one point. While this problem is intractable directly, in this joint work with Helge Ruddat and Bernd Siebert, we use toric dualities to translate the problem into a dual problem in a dual geometry. There the problem turns into a problem of computing period integrals, which we can readily solved via the techniques of Picard-Fuchs equations.

In my talk, I will limit to the case of X the projective plane and D a smooth conic. The before-mentioned toric dualities are the constructions of the Gross-Siebert programme. I hope to convey the observation that the dualities are natural and that the translation from counting problem to period integral is as well.

**Wednesday 9 November 2022, 3pm. Speaker: Beihui Yuan (Swansea)**

Title: 16 Betti diagrams of Gorenstein Calabi-Yau varieties and a Betti stratification of Quaternary Quartic Forms

Abstract: Motivated by the question of finding all possible projectively normal Calabi-Yau 3-folds in 7-dimensional projective spaces, we proved that there are 16 possible Betti diagrams for arithmetically Gorenstein ideals with regularity 4 and codimension 4. Among them, 8 Betti diagrams have been identified with those of Calabi-Yau 3-folds appeared in a list of 11 families founded by Coughlan-Golebiowski-Kapustka-Kapustka. Another 8 cannot be Betti diagrams of any smooth irreducible nondegenerate 3-fold. Based on the apolarity correspondence between Gorenstein ideals and homogeneous polynomials, and on our results on 16 Betti diagrams, we describe a stratification of the space of quartic forms in four variables.

This talk is based on the paper “Calabi-Yau threefolds in **P**^{n} and Gorenstein rings” by Hal Schenck, Mike Stillman and Beihui Yuan, and the preprint “Quaternary quartic forms and Gorenstein rings” by Michal and Grzegorz Kapustka, Kristian Ranestad, Hal Schenck, Mike Stillman and Beihui Yuan.

**Wednesday 16 November 2022, 3pm. Speaker: Ruijie Yang (Humboldt-Universität zu Berlin)**

Title: Zeroes of one forms and homologically trivial deformations

Abstract: In 1926, Hopf proved the Poincaré-Hopf theorem, which implies that if a compact differential manifold admits a nowhere vanishing vector field, then its topological Euler characteristic is zero. Dually, it is natural to ask the same question for one forms. In 1970, Tischler proved that the existence of a nowhere vanishing real closed one form induces a differentiable fiber bundle structure over the circle. In 2013, Kotschick conjectured that for compact Kähler manifolds, admitting a nowhere vanishing real closed one form is actually equivalent to the existence of a nowhere vanishing holomorphic one form. In this talk, I will show that Kotschick’s conjecture can be deduced from a conjecture of Bobadilla-Kollár on homologically trivial deformation. Therefore, Kotschick’s conjecture is true if the first Betti number of X is at least 2dim(X)-2 and the Albanese variety of X is simple. This is joint work with Stefan Schreieder.

**Wednesday 23 November 2022, 3pm. Speaker: Jonathan Lai (Imperial)**

Title: A Reconciliation of Mutations and Potentials

Abstract: Given a lattice polygon, one can consider the spanning fan to obtain a toric variety. A combinatorial mutation is an operation that takes one polygon to another, which induces a degeneration of one toric variety to the other. One can then attempt to study all toric degenerations of a fixed Fano variety through the study of polygons and their mutations. In another world, a set of algebraic tori can be glued together by birational maps, also called mutations, to form a cluster variety.

In this talk, I will explain a justification coming from mirror symmetry on why these two operations deserve to share the same name (in dimension 2). Given an orbifold del Pezzo surface X, there is a natural cluster variety Y that knows about the polytopes and mutations associated to X. Namely, there is a combinatorial object associated to Y called a scattering diagram, which is a collection of walls inside a vector space. The chambers, which correspond to tori in Y, are precisely the polygons coming from toric degenerations of X. This is based off ongoing joint work with Tim Magee and Ben Wormleighton.

**Wednesday 30 November 2022, 3pm. Speaker: Qaasim Shafi (Birmingham) Postponed to next term**

**Week 10: No seminar, because of the algebraic geometry event on Thursday/Friday**