2017 Term 2

Week 6, 14/02/2018

Oliver Anderson (University of Liverpool)

Introduction to h-topologies

The h-topology was first introduced in Voevodsky’s PhD thesis as part of an attempt of developing a homotopy theory of schemes, and has recently proven itself useful in the study of differential forms on singular varieties due to the efforts of Huber and her collaborators. In this talk I will give an introduction to the h-topology and some of its weaker cousins. More precisely I will start by giving their definitions, then provide some examples of coverings in these topologies and explain why they are not sub canonical. Then we will move on to refinement results and to conclude the talk mention some theorems concerning sheaves in the h-topologies.

Week 7, 21/02/2018

Sara Lamboglia (University of Warwick)

Tropical Fano Schemes

The classical Fano scheme of a variety X parametrizes linear spaces contained in X. It is well understood for certain classes of varieties such as toric varieties or hypersurfaces, but there are still many open questions concerning dimension and irreducibility. In this talk I am going to define the tropical analogue of the Fano scheme. I will show its relation with the classical Fano scheme and how it could help in the study of classical problems. Finally I will show some interesting examples where the tropical Fano scheme behaves differently from its classical counterpart.

Week 8, 28/02/2018

Erik Paemurru (Loughborough University)

Birational models of terminal sextic double solids

The rationality problem for algebraic varieties is one of the central problems in algebraic geometry. One way to prove irrationality is to show the variety is birationally rigid, which has far more insightful consequences than just irrationality. In this talk, I will discuss birational rigidity of sextic double solids with an isolated compound A_n singularity. In particular, I will introduce birational rigidity, analytic singularities, and discuss progress so far.

Week 9, 07/03/2018

Caitlin McAuley (University of Sheffield)

An introduction to Bridgeland stability conditions

The space of stability conditions is a complex manifold associated to a triangulated category. The definition of a stability condition was motivated by work in string theory and as such, an understanding of the stability manifold will have important consequences in mirror symmetry. Additionally, stability conditions play an increasingly interesting role in birational geometry.

I'll introduce stability conditions on an arbitrary triangulated category and discuss some of their most important features, as well as discussing examples of stability manifolds associated to the derived categories of projective varieties and quivers.

Week 10, 14/03/2018

Lawrence Barrott (University of Cambridge)

Chow Theory for log varieties

Log geometry keeps appearing in mirror symmetry, but the geometric behaviour of these objects is not always clear. In this talk we extend a classic invariant to this new setting.

2017 Term 3

Week 2, 30/04/2018

Ben Anthes (Philipps-Universität Marburg)

Gorenstein stable surfaces with K^2=2 and chi=4

I will explain the basic ideas underlying work in progress about a stratification of the moduli space of Gorenstein stable surfaces with K^2 = 2 and holomorphic Euler characteristic equal to 4. Stable surfaces are the kind of surfaces occurring as the degenerations of (canonical) surfaces of general type, giving rise to the KSBA-compactification of the moduli space. The components under consideration are stratified by means of the number and type of non-canonical singularities, which we study by using an explicit isomorphism between the moduli space with a moduli space of certain plane curves. Considering the corresponding stratification of the moduli space of plane curves, one can make make use of computer algebra systems to compute, e.g., the dimensions of (most of) the strata. We will conclude the talk with (drawings of) some interesting examples.

Week 3, 09/05/2018

Sjoerd Beentjes (University of Edinburgh)

Counting curves on Calabi-Yau threefolds via sheaves

By now the enumerative geometry of curves on Calabi-Yau threefolds has become a rich subject, partly thanks to input from string theory. There are different ways of 'counting’ curves and there is a web of conjectures (and theorems!) relating them. In this talk, I’ll present two famous ways of counting curves via sheaf-theoretic methods: Donaldson-Thomas (DT) invariants and Pandharipande-Thomas (PT) invariants.

After presenting some examples, I’ll discuss an approach to proving a comparison theorem between DT and PT invariants, due to Toda and Bridgeland. This uses wall-crossing methods and the motivic Hall algebra. If time permits, I’ll sketch the statement of another such comparison result, the crepant resolution conjecture for Donaldson-Thomas invariants, the proof of which is joint work with J. Calabrese and J. Rennemo.

Week 4, 14/05/2018

Joshua Jackson (University of Oxford)

Non-reductive Geometric Invariant Theory and Applications

Geometric Invariant Theory, which one may characterise as 'the art of quotienting algebraic varieties by group actions', has long been a central tool in algebraic geometry. In particular, it is of tremendous use in the construction and study of moduli spaces: perhaps most notably the Deligne-Mumford moduli space of stable curves, and the moduli space of semistable coherent sheaves over a projective scheme. Less well known, however, is the recent generalisation of GIT to actions of non-reductive groups, due to Berczi-Doran-Hawes-Kirwan. I will attempt to explain why non-reductive GIT is much harder, what results are known about it, and some of the cool things we can do with it - including joint work generalising the two moduli spaces mentioned above.

Week 5, 21/05/2018

Riccardo Moschetti (Universitetet i Stavanger)

On coherent sheaves of small length on the affine plane

I would like to discuss some relations between the problem of classifying modules of finite length over k[x,y] and motivic Donaldson-Thomas invariants. In particular, I will compare our result for length lesser or equal than four with the motivic class of the moduli stack parametrizing such modules. This is a joint work with Andrea Ricolfi.

Week 6, 29/05/2018

Elena Berardini (Aix-Marseille Unviersité)

Codes on Algebraic Surfaces

In this talk we will discuss algebraic codes built starting from abelian surfaces defined over finite fields. Our goal will be to build a code over an abelian surface defined over a finite field, e.g. the jacobian of a genus 2 curve, then to characterize this code by giving its length, dimension and an upper bound for its minimal distance. In order to do so, we will introduce intersection theory over surfaces, will use some known results of algebraic geometry over surfaces and we will look at some theory regarding rational points of irreducible curves over an abelian surface.

Week 7, 04/06/2018

Sara Torelli (Leibniz Universität Hannover)

On the unitary flat bundle of the second Fujita decomposition of a semistable fibration of curves

The second Fujita decomposition asserts that the Hodge bundle of a semistable fibration of curves decomposes into a direct sum of a unitary flat bundle and an ample bundle. Moreover, such decomposition is defined by the geometric variation of the Hodge structure and the unitary flat summand is nothing but its locally constant part. In the seminar I will summarize some results obtained on the rank and on the monodromy of this bundle in case it admits an extra-property related to the geometry of the total space and I will discuss some applications. The results are in collaboration with Víctor González-Alonso, Gian Pietro Pirola and Lidia Stoppino.

Week 8, 11/06/2018

Annalisa Grossi (Alma Mater Studiorum, Bologna)

Automorphisms of O’Grady-six type manifolds acting non-trivially on cohomology

I will talk about automorphisms of a specific six-dimensional example of Hyperkähler manifolds: O’Grady six-dimensional example (OG6). I will focus on finite and non-symplectic automorphisms and I will explain a first result about existence and uniqueness of a specific class. The main goal is to study the image of the map ν : Aut(X) → O(H^2(X, Z)), where X ∼ OG6. In order to do this I need some tools of lattice theory which are useful to understand the action of a finite order isometry ϕ on H^2(X, Z) and to find invariant and co-invariant lattices with respect to the action of ϕ. I will show some properties that an isometry of H^2(X, Z) needs to have in order to be in the image of ν. This is sufficient to say that this isometry comes from an automorphism of X.

Week 9, 18/06/2018

Alex Torzewski (University of Warwick)

Motives over Shimura Varieties

Motives are intended as an intermediate object between varieties and cohomology. They should behave in a way similar to cohomology and yet admit a purely geometric definition. We shall sketch the basic properties of classical motives and outline how several important conjectures fit into this motivic framework. Time permitting we shall also discuss some new results on motives over Shimura varieties.

Week 10, 25/06/2018

Matilde Manzaroli (École Polytechnique)

Construction of real algebraic curves on the quadric ellipsoid

The classification of topological types, up to homeomorphism, realized by a real algebraic curve of fixed degree in the real projective plane is a classical subject in which huge progresses have been made since 1970. In this talk, I would like to present a similar classification in a different ambient surface: the quadric ellipsoïd, X. I will explain how to do a classification of the topological types realized by real algebraic curves of bidegree (d, d) on X, in particular giving the main idea of one possible construction of such curves for d=5.