Junior Algebraic geometry Warwick Seminar (JAWS)

Seminar Organisers: Tommaso Faustini, Mayo Mayo García and Marc Truter

Rooms and Times

Term 1: D1.07 3 - 4 pm Thursdays

Term 2: D1.07 3 - 4 pm Thursdays

Term 3: MS.04 3 - 4 pm Thursdays

Week 2 Jan 22

Sae Koyama (Cambridge)

Enumerating elliptic curves in toric threefolds using tropical geometry

Tropical methods have been a powerful tool in enumerative geometry. The main idea is that we can define “tropical” objects, which are combinatorial in nature, and use those to define tropical invariants which will coincide with algebraic invariants. For instance, Mikhalkin showed that genus 0 curve counts coincide with weighted counts of genus 0 tropical curves. The difficulty with extending this to higher genus is the existence of tropical objects that don’t come from algebraic ones. We discuss how logarithmic techniques can be used to deal with these contributions and derive enumerative consequences in the case of elliptic curves in toric threefolds.

Week 5 Feb 12

Sean Fitzgerald (Trinity)

Exploring the combinatorics of Hurwitz numbers

Hurwitz numbers, introduced by Hurwitz in the late 19^th century, enumerate branched morphisms with fixed ramification data. In recent years these numbers have proven to be closely related to the likes of tropical geometry, Gromov-Witten theory and topological recursion, with many new variants appearing in the literature. Over the course of this talk we will discuss the basic theory of these counts and their variants, before exploring in more detail both the single and pruned Hurwitz numbers. We will employ a combinatorial/graph-theoretic approach, using the likes of branching graphs, Hurwitz mobiles and lattice paths.

Week 6 Feb 19

Siao Chi Mok (Cambridge)

Compactifications and degenerations of configuration spaces, via tropical geometry

Given some tropical moduli data, I will outline a procedure to construct a degeneration of a toric variety. These degenerations can be used to describe well-behaved compactifications of moduli spaces. This will all be illustrated in the context of logarithmic Fulton—MacPherson configuration spaces, which compactify configuration spaces of X\D, where X is a toric variety and D is its toric boundary. These constructions can be generalised to the case where X is a smooth projective variety with a simple normal crossings divisor D. We will also see how these constructions can be adapted to construct a log smooth degeneration of the Fulton—MacPherson configuration space of a smooth projective variety Y.

Week 7 Feb 26

Tianchen Zhao (Exeter)

Good Reduction of Generalized Kummer surfaces in the non-supersingular case.

In this talk, we study the good reduction of generalized Kummer surfaces, which are K3 surfaces obtained as minimal resolutions of quotients of abelian surfaces by finite groups. This extends the result of Lazda and Skorobogatov on Kummer the abelian surface has non-supersingular reduction and the group is cyclic. This extends the result of Lazda and Skorobogatov on Kummer surfaces.

Week 8 Mar 5

Yijue Hu (Nottingham)

K-stability of smooth Fano complete intersections

The study of K-stability originates from the search for Kähler-Einstein metrics and is essential to the construction of moduli spaces of Fano varieties. In this talk, I will review known results on the K-stability of smooth Fano hypersurfaces and higher-codimension varieties, and introduce a powerful tool for determining K-stability known as the Abban-Zhuang method. Then I will show how this method is used to prove the K-stability of codimesion two smooth complete intersections of hypersurfaces of degree 2 and n in \mathbb{P}^{n+2}.

Week 9 Mar 12

Week 10 Mar 19

Filippo Papallo (Genova)

On the weak Brill-Noether problem for Enriques surfaces

The weak Brill–Noether problem for stable vector bundles concerns the cohomology vanishing of a general sheaf on a fixed variety. While Brill–Noether theory is well understood for curves, its surface analogue remains nowadays far less developed. The aim of the talk is to outline two complementary path towards this question: one using Bridgeland stability and wall‑crossing, and another more computational one, based on explicit projective models of Enriques surfaces.

Week 1 Apr 30

Week 2 May 7

Week 3 May 14

Week 4 May 21

Matthias Ferreira (Sorbonne)

Week 5 May 28

Week 6 Jun 4

Week 7 Jun 11

Brais Gerpe Vilas (Sheffield)

Week 8 Jun 18

Week 9 Jun 25

Week 10 Jul 2

Week 3 Oct 23

Daniel Green Tripp (Bristol)

An equidistribution theorem on toric varieties

A version of Weyl’s equidistribution theorem states that, for almost all points in the n-dimensional compact torus, the average of the Dirac measures on the powers of the point converges to the normalised Haar measure. We will explore how to generalise this statement to the n-dimensional complex algebraic torus; at this moment, toric varieties will emerge as our saviours, giving us the convergence. Time permitting, I will also discuss possible future directions on finding even more interesting convergences. Based on an ongoing joint work with Farhad Babaee.

Week 5 Nov 6

Parth Shimpi (Glasgow)

It’s supported on P1, how complicated could it be?

"Not complicated at all", at least that is the case I will make. In the deep dark forest prowled by serpentine complexes and monstrous sheaves that are challenging to identify (let alone analyse), rational curves provide a warm and cozy retreat --- the calm of knowing where everything sits and nothing can go wrong. Be it a surface, a threefold, or simply a single curve: if all you care about is what sits near a P1, I will sketch how a little representation theory goes a long way in providing a complete map of the derived category.

Week 7 Nov 20

Ines Chung-Halpern (LSGNT)

Cluster varieties, integral linear manifolds and mirror symmetry

Cluster varieties are a class of algebraic varieties carrying deep combinatorial structure, which allows us to adapt methods from toric geometry to a more general setting. In this talk, I will introduce cluster varieties in dimension 2 and their tropicalisations, which combinatorially encodes much of their geometric data in its integral linear structure. We will see how the analogue of polytopes on these linear manifolds gives rich examples of compactifications of cluster varieties, and allows us to compute examples of mirror symmetry for Fano varieties.

Week 8 Nov 27

Dan Simms (LSGNT) [Time and Room Change: 1 - 2 pm B3.01]

Localisation in Enumerative Geometry

This talk will introduce the modern approach to studying enumerative geometry via the intersection theory of moduli spaces. When the moduli spaces are poorly behaved, one must jump through several hoops to find a ‘virtual fundamental class’ in order to do this properly. Rather than going down this road, we will see how things can be simplified in the presence of a torus action. This will take us on a journey through equivariant cohomology, and will hopefully end with a couple of nice applications.

COW Seminar (Warwick) 2 - 5pm

Week 8 Nov 28

CALF Seminar (Warwick, IAS seminar room) 1 - 5pm