Junior Algebraic geometry Warwick Seminar (JAWS)

Seminar Organisers: Marc Truter and David Hubbard

Reading groups:

Term 1 (GIT) organised by Marc Truter: The term 1 reading group will be on GIT, following Victoria Hoskins' notes. See my webpage for the notes from our talks.

Term 2 (K-Stability) organised by Marc Truter: The term 2 reading group will be on K-stability, following Harold Blum's notes

Rooms and Times

Term 1: D1.07 12-1pm Thursdays for reading group, D1.07 3-4pm Thursday for talks

Term 2: B1.12 12-1pm Thursdays for reading group, D1.07 3-4pm Thursdays (Week 7: 11-12 A1.01)

Term 3: MB0.08 3-4pm Thursdays

Term 2

Week 1 Jan 9

Reading group [Peize Liu, GIT and blowups](B1.12 12-1pm Thursday)

Week 3 Jan 23

Reading group [Marc Truter, Introduction to K-stability](B1.12 12-1pm Thursday)

Arun Soor - Oxford (D1.07 3-4pm Thursday)

Six-functor formalisms and derived rigid-analytic geometry

The “yoga” of the six operations is a notion which goes back to Grothendieck. It was originally an evocative notion alluding to the functorial properties satisfied by various categories of coefficients on geometric objects X, for instance D-modules or l-adic sheaves.

In recent years, due to the breakthrough works of Liu--Zheng, Gaitsgory--Rozenblyum, and Mann, our understanding of the six operations has developed from a “yoga" to a rigorously defined notion. I will try to explain this development.

Then I will try to explain how to construct a six-functor formalism for quasi-coherent sheaves in derived rigid-analytic geometry, using the recent work of Ben-Bassat--Kelly--Kremnitzer. Along the way, I'll try to explain why it's quite natural to derive rigid-analytic geometry for this and other reasons.

Week 4 Jan 30

Reading group [Marc Truter, K-stability of polarised varieties](B1.12 12-1pm Thursday)

Week 5 Feb 6

Reading group [Alvaro Gonzalez Hernandez, Odaka's theorems](B1.12 12-1pm Thursday)

Week 6 Feb 13

Reading group [Tommaso Faustini, Valuations and Test configurations](B1.12 12-1pm Thursday)

Amy Li - Texas Austin (D1.07 3-4pm Thursday)

Week 7 Feb 20

Reading group [CANCELLED DUE TO COW]

Ruth Wye - Bath [A1.01 11-12 (Rescheduled due to COW)]

Week 8 Feb 27

Reading group [TBD, K-stability of Fanos and Valuations](B1.12 12-1pm Thursday)

Week 9 Mar 6

Reading group [TBD, K-stability of Fanos and anti-canonical divisors](B1.12 12-1pm Thursday)

Week 10 Mar 13

Reading group [TBD, K-moduli of Fano varieties](B1.12 12-1pm Thursday)

Joris Köfler - Max Planck (Leipzig) (D1.07 3-4pm Thursday)

Term 1

Week 2 Oct 10

Reading group [Marc Truter, Introduction to Moduli problems and GIT] (2-3pm Thursday C1.06)

Week 3 Oct 17

Reading group [Arnaud Vilpert, Moduli problems] (12-1pm Thursday D1.07)

Week 4 Oct 24

Reading group [Chunkai Xu, Groups and actions] (12-1pm Thursday D1.07)

Joe Malbon - Edinburgh (3-4pm Thursday D1.07)

Classification of Algebraic Varieties

Classification - the identification of similar objects and the distinction of different ones, as well as the construction of spaces that parametrise such equivalence classes - is a guiding meta-principle in algebraic geometry. It was realised around the turn of the last century that isomorphisms are too rigid a notion of equivalence for classification to be achievable, and that the more flexible notion of birational equivalence should be used instead. This point of view was successfully applied to algebraic surfaces, whose birational classification was more or less completed by the 1950s.

For higher-dimensional varieties, it is often said that all varieties are birationally (and conjecturally) constructed from Fano, Calabi-Yau, and canonically polarised varieties, and thus classification may proceed inductively based on dimension. In this talk, I will explain the birational classification of varieties from the viewpoint of the minimal model program, which is an algorithm that constructs varieties in this inductive way. If time permits, I will explain the application of the minimal model program to the construction of moduli spaces, which parametrise equivalence classes of varieties with similar properties, and illustrate some of my own work on the construction of the moduli space of a particular family of K-polystable Fano threefolds.

Week 5 Oct 31

Reading group [Alvaro Gonzalez Hernandez, Affine GIT](12-1pm Thursday D1.07)

Week 6 Nov 7

Reading group [Tommaso Faustini, Projective GIT] (12-1pm Thursday D1.07)

Thamarai Valli - UCL (LSGNT) (3-4pm Thursday D1.07)

Complete intersections in Toric varieties: A Combinatorial Adventure!

Toric varieties are a fascinating class of algebraic varieties with a rich combinatorial structure that simplifies many computations. Seeking to extend this combinatorial advantage to broader classes, we focus on complete intersections within toric varieties, which inherit some of the combinatorial features of toric varieties. This additional structure makes them particularly useful for the classification of Fano varieties, a central problem in algebraic geometry.

In this talk, we’ll begin with a formal introduction to toric varieties as GIT quotients, then explore complete intersections within these varieties and their associated combinatorial properties. We will then examine how these properties aid in classifying Fano varieties. If time permits, I’ll also discuss my recent work on the classification of Fano complete intersections in toric varieties.

Week 7 Nov 14

Reading group [Marc Truter, Stability] (12-1pm Thursday D1.07)

Week 8 Nov 21

Reading group [Peize Liu, Moduli of vectors bundles I] (12-1pm Thursday D1.07)

Michela Barbieri - Imperial (LSGNT) (3-4pm Thursday D1.07)

Derived Categories and Mirror Symmetry Conjecture

Derived categories were originally developed as a homological algebra tool, but in fact are very interesting from a geometric point of view too. One example of this is Homological Mirror Symmetry (HMS), which comes from a mysterious and poorly understood relationship between complex and symplectic geometry, and originates from physics, specifically string theory. Generally, the HMS conjecture says that given some 'complex geometry' X, there is a mirror 'symplectic geometry' Y such that the derived category of coherent sheaves on X, denoted D^b(Coh X), is equivalent to a Fukaya category of Y, denoted Fuk(Y). In this talk I'll talk about toric HMS from the geometric invariant theory (GIT) perspective - this story is nice because it's well-understood and very combinatorial. No prior knowledge of derived categories, symplectic geometry, GIT, etc will be assumed. One of the main goals of this talk will be to explain what derived categories are informally, with the HMS stuff hopefully giving a motivation for studying them.

Week 9 Nov 28

Reading group [Chunkai Xu, Moduli of vectors bundles II] (12-1pm Thursday D1.07)

Week 10 Dec 5

Reading group [Peize Liu, Hypersurfaces and blowups] (12-1pm Thursday D1.07)

Xuanchun Lu - Cambridge (3-4pm Thursday D1.07)

Double Ramification cycles in the punctured setting

The projective line $\mathbb{P}^1$ is the only algebraic curve on which every degree 0 divisor is principal. For any other algebraic curve, it is thus natural to ask when a 'random' divisor on the said curve becomes principal.

Generalising classical works of Abel--Jacobi, a solution to the above problem may be given as a collection of cycles on the moduli spaces $M_{g, n}$ of genus g, n-marked algebraic curves. These cycles are known as the double ramification (DR) cycles. Since $M_{g, n}$ is not compact, these cycles do not fully capture the geometry of our situation. We thus need to seek an extension of DR cycles to some compactification of $M_{g, n}$.

In this talk, I will explain some of the challenges involved in constructing such an extension, and how they can be resolved using ideas and techniques from logarithmic geometry. I will also highlight some downstream enumerative applications of DR cycles. Time permitting, I will attempt to shed some light on the appearance of the word 'punctured' in the title.