Warwick Algebraic Geometry Seminar

Abstracts

Jenia Tevelev (University of Massachusetts at Amherst) - Derived Category of the Moduli Space of Stable Rational Curves

I will discuss work in progress with Ana-Maria Castravet verifying a surprising conjecture of Orlov and Kuznetsov on the equivariant structure of the derived category (and even K-theory) of the moduli space of stable rational curves.

Alan Thompson (University of Warwick) - Towards a Compactification of the Moduli Space of K3 Surfaces of Degree 2

Ever since moduli spaces of polarised K3 surfaces were constructed in the 1980's, people have wondered about the question of compactification: can one make the moduli space of K3 surfaces compact by adding in some boundary components in a "nice" way? Ideally, one hopes to find a compactification that is both explicit and geometric (in the sense that the boundary components provide moduli for degenerate K3's). I will present on joint work in progress with V. Alexeev, which aims to solve the compactification problem for the moduli space of K3 surfaces of degree 2.

Michel van Garrel (KIAS) - From Local to Relative Gromov-Witten Invariants via Log Geometry

Let X be a smooth projective variety and let L be a line bundle corresponding to a smooth ample divisor D. One may wonder if counting curves in X twisted by L has any bearing on counting log curves of (X,D). In this joint work with Graber and Ruddat, we give a positive answer to that question and show that the genus zero local Gromov-Witten invariants of L are maximally tangent relative Gromov-Witten invariants of the pair (X,D). This generalizes an old formula of Takahashi. The key technical ingredient is the theory of log stable maps by Gross and Siebert.

Chunyi Li (University of Edinburgh) - Bridgeland Stability Conditions on Fano Threefolds

The notion of a stability condition on a triangulated category has been introduced by Bridgeland around fifteen years ago. The existence of geometric stability conditions on threefolds has been a core open problem of the field. The existence of such conditions is equivalent to some generalized version of Bogomolov inequalities and will imply new bounds on Chern classes of stable sheaves (in particular, ideal sheaves of sub-varieties). I’ll review the definition of stability conditions and explain the importance and difficulty of the existence problems. I’ll conclude with a 'proof' for the case of Fano threefolds.

Stavros Papadakis (University of Ioannina) - Weak Lefschetz Property and Stellar Subdivisions of Simplicial Complexes

Assume sigma is a face of a Gorenstein* simplicial complex D. The talk, which is based on joint work with Janko Boehm (Kaiserslautern), will be about the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring k[D] of D is equivalent to the same property of the Stanley-Reisner ring k[D_sigma] of the stellar subdivision D_sigma.

Toby Stafford (University of Manchester) - Noncommutative Rational Surfaces

One of the major open problems in non-commutative algebraic geometry is the classification of non-commutative surfaces (or of connected graded algebras of Gelfand-Kirillov dimension 3). Artin has conjectured that the corresponding division rings are known, with the generic case being the ring of fractions of the so-called Sklyanin algebra. In this talk we will discuss progress in classifying the non commutative surfaces birational to Proj of that algebra. In particular, non-commutative analogues of blowing up and down are understood, and this has for example been used to determine the subalgebras of the Sklyanin algebra.

This talk will survey this subject and show in particular that Van den Bergh’s quadric surfaces are minimal models in a very strong sense.

This is joint work with Dan Rogalski and Sue Sierra.

Ingo Blechschmidt (Universität Augsburg) - First Steps in Synthetic Algebraic Geometry

We describe how the internal language of certain toposes, the associated little and big Zariski toposes of a scheme, can be used to give simpler definitions and more conceptual proofs of the basic notions and observations in algebraic geometry.

The starting point is that, from the "internal point of view" of the little Zariski topos, sheaves of rings and sheaves of modules look just like plain rings and plain modules. In this way, some concepts and statements of scheme theory can be reduced to concepts and statements of intuitionistic linear algebra. This simplifies working with sheaves and brings conceptual clarity.

The internal language of the big Zariski topos goes even further. It incorporates Grothendieck's functor-of-points philosophy in order to cast modern algebraic geometry, relative to an arbitrary base scheme, in a naive language reminiscient of the classical Italian school. The base scheme looks like the one-element set from this point of view.

The talk gives an introduction to this topos-theoretic point of view of algebraic geometry in the hope that it's useful to working algebraic geometers. No prior knowledge about toposes is supposed.

Alastair Craw (University of Bath) - Multigraded Linear Series

I'll describe joint work with Yukari Ito and Joseph Karmazyn where we provide the natural moduli space description of the minimal resolution Y of a surface singularity A^2/G for any finite subgroup G in GL(2). In fact, our results apply to any subminimal partial resolution. The approach blends geometry and algebra by constructing every such space as the multigraded linear series of a vector bundle on Y.

Andrew MacPherson (University of Tokyo) - Fibre Products and Derived Geometry

Derived geometry can be defined as the universal way to attach fibre products to a category of manifolds (or smooth algebraic varieties) compatibly with Cartesian products and "glueing". I'll talk about some simple examples and consequences of the definition - such as the preservation of transverse fibre products in the differentiable setting and the existence of a good virtual dimension invariant.

Victoria Hoskins (Freie Universität Berlin) - Group Actions on Quiver Varieties and Applications

We consider two types of actions on moduli spaces of quiver representations over a field k and we decompose their fixed loci using group cohomology. First, for a perfect field k, we study the action of the absolute Galois group of k on the points of this quiver moduli space valued in an algebraic closure of k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by the Brauer group of k. Second, we study algebraic actions of finite groups of quiver automorphisms on these moduli spaces; the fixed locus is decomposed using group cohomology and each component has a modular interpretation. As an application, we describe the symplectic and holomorphic geometry of these fixed loci in hyperkaehler quiver varieties. This is joint work with Florent Schaffhauser.

Barbara Bolognese (University of Sheffield) - Nef Cones of Hilbert Schemes of Points via Bridgeland Stability

Carrying out the Minimal Model Program for moduli spaces is a classical and extremely challenging problem. In this talk, we will deal with a particular moduli space, namely the Hilbert scheme of points on a surface with irregularity zero. After explaining the connection between the birational models of a variety and the combinatorics of its Nef cone, we will show how Bridgeland stability conditions are a powerful machinery to produce extremal rays in the Nef cone of the Hilbert scheme. Time permitting, we will give a complete description of the Nef cone in some examples of low Picard rank. This is joint work with J. Huizenga, Y. Lin, E.Riedl, B. Schmidt, M. Woolf and X. Zhao.

Juan Garza Ledesma (University of Warwick) - Graded Ring Constructions: Some New Examples

In recent times, some people have been re-studying minimal surfaces of general type with K²=7 and p_g=4 (they've been studied at least since Enriques' times). In particular, it's interesting to understand and construct the possible deformations relating the subfamilies of such surfaces. In this talk I'll explain how to approach the problem via Miles Reid's graded ring methods, in particular I'll give an example involving explicit constructions of Gorenstein rings of codim 6, 4 and 3 and their deformation families.

Tom Ducat (RIMS, Kyoto University) - Constructing Q-Fano 3-folds following Prokhorov & Reid

Prokhorov & Reid constructed two families of Q-Fano 3-folds of index 2 by calculating a simple kind of Sarkisov link which blows up a cyclic quotient singularity and then contracts a divisor to a singular curve. I'll explain a generalisation of this construction which gives five more families; four of index 2 and one of index 3. Two families have the same Hilbert series and I'll explain how to construct them as codimension 4 Tom and Jerry unprojections.

Stefan Kebekus (Albert-Ludwigs-Universität Freiburg) - Higgs Sheaves on Singular Spaces, the Nonabelian Hodge Correspondence, and the Miyaoka-Yau Inequality for Minimal Varieties of General Type

We establish the Miyaoka-Yau inequality for the tangent sheaf of any minimal, complex, projective variety X of general type, with only klt singularities. In the case of equality, we prove that the canonical model of X has only quotient singularities and is uniformized by the unit ball. This is joint with Greb, Peternell, Taji.

Ulrich Derenthal (Leibniz Universität Hannover) - Cox Rings over Nonclosed Fields

For a wide class of varieties over algebraically closed fields, Cox rings were defined and studied by Cox, Hu, Keel, Hausen, Hassett and others. We give a new definition of Cox rings for suitable varieties over arbitrary fields that is compatible with universal torsors, which were introduced by Colliot-Thélène and Sansuc. We study their existence and classification, and we make their relation to universal torsors precise. This is joint work with Marta Pieropan.

Imran Qureshi (Lahore University of Management Sciences) - Fano 4-folds in Gorenstein Formats

The classification of Fano varieties is an interesting problem, as the number of deformation families of Fano manifolds in each dimension has been proved to be finite by Kollar and Mori. The classification of Fano manifolds in dimension less than or equal to 3 is complete. In dimension 4, the classification is complete in the case of Fano index greater than 1. The aim of this talk is to present some new deformation families of 4-dimensional Fano manifolds of index 1, constructed as complete intersections in some ambient Gorenstein key varieties (formats). I will also describe an algorithmic approach to find all possible isolated orbifolds with fixed canonical class in a given fixed Gorenstein key variety. (This is joint work in progress with Gavin Brown and Alexander Kasprzyk.)