Junior Algebraic geometry Warwick Seminar (JAWS)

Rooms and Times

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

Term 2: D1.07 3-4pm Thursdays

Term 3: MB0.08 3-4pm Thursdays

Term 1

The reading group this term will be on GIT, following Victoria Hoskins notes. See my webpage for the notes from our talks.

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 (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 (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 (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 (3-4pm Thursday D1.07)