# AGMP seminar 2022-23

## A tasting menu in non-reductive GIT

### Thursday 20 October 2022

**Speaker:** Josh Jackson (Sheffield)**Location:** J11**Time:** 2.00pm

**Abstract:** Non-reductive GIT is a new generalisation of Mumford's classical GIT, that allows quotienting algebraic varieties by actions of non-reductive groups. I will explain how it works, and indicate the range of applications it has seen so far.

## Curve counting on surfaces and topological strings

### Thursday 3 November 2022

**Speaker:** Andrea Brini (Sheffield)**Location:** J11**Time:** 2.00pm

**Abstract:** I will describe some correspondences relating different types of enumerative invariants associated to a pair (X,D), with X a complex projective surface and D a singular anticanonical divisor in it. These include the log Gromov-Witten invariants of the pair, the Gromov-Witten invariants of an associated higher dimensional Calabi-Yau variety, the open Gromov-Witten invariants of certain special Lagrangians in toric Calabi--Yau threefolds, the Donaldson-Thomas theory of a class of symmetric quivers, and certain open and closed Gopakumar-Vafa-type invariants. A lot of my personal motivation to believe in the correspondences comes from dualities in physics, but the talk won't require any knowledge of string theory, and nobody will get hurt as a result. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants. Based on past&present work with Bousseau, van Garrel, Nabijou, and our very own Yannik.

## Stability conditions with massless objects

### Thursday 10 November 2022

**Speaker:** Jon Woolf (Liverpool)**Location:** J11**Time:** 2.00pm

**Abstract:** The Bridgeland stability space of a triangulated category is a non-compact complex manifold with a wall-and-chamber structure capturing interesting aspects of the category’s structure. I will describe joint work with Broomhead, Pauksztello and Ploog in which we partially compactify the stability space by allowing `degenerate’ stability conditions with massless objects. One reason this is interesting is that the added boundary points are closely related to the walls. I will illustrate this connection in low-dimensional examples arising from quivers with two vertices.

## The logarithmic Hilbert scheme

### Thursday 17 November 2022

**Speaker:** Patrick Kennedy-Hunt (Cambridge)**Location:** J11**Time:** 2.00pm

**Abstract:** The Hilbert scheme of a projective variety X is the moduli space of closed subschemes of X. In this talk we will discuss a version of the Hilbert scheme for a pair (X,D) with D a (reasonable) divisor on X. This logarithmic Hilbert scheme is a special case of the logarithmic Quot scheme. As motivation, note hard algebraic geometry problems can be studied by degenerating to simpler situations. Logarithmic geometry provides a suite of tools to study such degenerations. For example, these techniques have been applied to study (logarithmic) GromovWitten theory and more recently (logarithmic) Donaldson-Thomas theory. The logarithmic Hilbert scheme of curves on a threefold is the moduli space studied in logarithmic Donaldson-Thomas theory. A long-term hope is to study other moduli spaces of coherent sheaves with techniques from logarithmic geometry.

## **** Cancelled ****

### Thursday 24 November 2022

**Speaker:** Omar Kidwai (Birmingham)**Location:** J11**Time:** 2.00pm

**Abstract:** ...

## Cross-ratios and perfect matchings

### Thursday 1 December 2022

**Speaker: **Rob Silversmith (Warwick)**Location:** J11**Time:** 2.00pm

**Abstract:** Given a bipartite graph G (subject to a constraint), the "cross-ratio degree" of G is a non-negative integer invariant of G, defined via a simple counting problem in algebraic geometry. I will discuss some natural contexts in which cross-ratio degrees arise. I will then present a perhaps-surprising upper bound on cross-ratio degrees in terms of counting perfect matchings — the proof involves Gromov-Witten theory. Finally, I will discuss the tropical side of the story.

## WDVV and commutativity equations, and their rational and trigonometric solutions

### Thursday 12 January 2023

**Speaker:** Misha Feigin (Glasgow)**Location:** J11**Time:** 2.00pm

**Abstract:** In the theory of WDVV equations $F_i G^{-1} F_j = F_j G^{-1} F_i$ the constant matrix $G$ is usually assumed to be a linear combination of the matrices $F_i$ of the third order derivatives of the prepotential $F$. I would like to explain that this assumption follows from the equations under a non-degeneracy condition. Then, these equations admit a big class of rational solutions determined by configurations of vectors known as V-systems which include root systems. In the trigonometric settings the situation is much more restrictive, and known examples are given by non-simply-laced root systems and their projections.

## **** Cancelled ****

### Thursday 19 January 2023

**Speaker:** Omar Kidwai (Birmingham)**Location:** J11**Time:** 2.00pm

**Abstract:** ...

## Deformations of hypersurfaces with non-constant Alexander polynomial

### Thursday 9 February 2023

**Speaker:** Remke Kloostermann (Padua)**Location:** J11**Time:** 2.00pm

**Abstract:** Let *X* be an irreducible hypersurface in *P**n* of degree *d*. Suppose *X* has at most isolated singularities. Then *h**i*(*X*)=*h**i*(*P**n*) holds for *i*∉{*n*−1,*n*,2*n*−2}. Smooth hypersurfaces and most hypersurfaces with isolated singularities satisfy the equality *h**n*(*X*)=*h**n*(*P**n*). In this talk we consider the case where *h**n*(*X*)>*h**n*(*P**n*), i.e., hypersurfaces with defect. Moreover, we will limit ourselves to hypersurfaces with at most semi- weighted homogeneous (e.g., ordinary multiple points or ADE-singularities). We show that if (*d*,*n*) is not in an explicit finite list then the equianalytic deformation space of *X* is not *T*-smooth, i.e., this space is nonreduced or its dimension is larger than expected. A similar statement holds true for *X* if the *d*-fold cover *Y* of *P**n* ramified along *X* satisfies *h**n*+1(*Y*)>*h**n*+1(*P**n*+1). This latter result generalizes classical examples of B. Segre of degree 6*m* curves in *P*2 with 6*m*2, 7*m*2, 8*m*2 and 9*m*2 cusps and deformation space larger than expected.