Algebraic Geometry Seminar 22/23 Term 3
The algebraic geometry seminar in Term 3 2022/2023 will usually meet on Wednesdays at 3pm in MS.03, though we may sometimes change to allow speakers from other time zones.
See the talks from last term here.
Wednesday 26 April 2023, 3pm. No seminar.
Wednesday 3 May 2023, 3pm. Speaker: Miles Reid (Warwick)
Title: Cayley cubics and Enriques sextics
Abstract: A Cayley cubic is a cubic surface of PP^3 with nodes at the 4 coordinate points. It automatically passes through the 6 coordinate lines. An Enriques sextic is a sextic surface that passes doubly through the 6 coordinate lines. Putting both of these linear systems together into a graded ring gives a toric extraction of the 6 coordinate lines.
This construction leads naturally to the Enriques-Fano variety that is the toric variety (PP^1 x PP^1 x PP^1)/(±1) embedded in PP^13 by monomials corresponding to the face-centred cube. It can also be seen as the midpoint of the standard monoidal transformation PP^3 - -> PP^3 given by (x,y,z,t) |-> (1/x,1/y,1/z,1/t).
Wednesday 10 May 2023, 3pm. No seminar
Wednesday 17 May 2023, 3pm. Speaker: Andrew Kresch (University of Zurich)
Title: On rationality in families and equivariant birational geometry
Abstract: In this talk I will recall some developments, connected with rationality in families of varieties, and explain some analogous developments in equivariant birational geometry, obtained in recent joint work with Brendan Hassett and Yuri Tschinkel.
Wednesday 24 May 2023, 3pm. MS.04 -- note unusual room! Speaker: Hal Schenck (Auburn University)
Title: Calabi-Yau threefolds in P^n and Gorenstein rings
Abstract: A projectively normal Calabi-Yau threefold X in P^n has an ideal I_X which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such CY threefolds have been extensively studied when I_X is a complete intersection, as well as in the case where X is codimension three; in both these cases the algebra is well understood. We study the situation in codimension four or more, by lifting Artinian Gorenstein ideals obtained from Macaulay's inverse systems. This leads to the construction of CY threefolds with Hodge numbers not previously known to appear. (joint work with M. Stillman and B. Yuan).
Wednesday 31 May 2023, 3pm. Speaker: Milena Hering (Edinburgh)
Title: Stability of Toric Tangent bundles
Abstract: In this talk I will give a brief introduction to slope stability and present a combinatorial criterion for the tangent bundle on a polarised toric variety to be stable in terms of the lattice polytope corresponding to the polarisation. I will then present a theorem that on toric surfaces and toric varieties of Picard rank 2 there exists an ample line bundle such that the tangent bundle is stable if and only if the underlying variety is an iterated blow up of projective space. This is joint work with Benjamin Nill and Hendrik Süss.
Wednesday 7 June 2023, 3pm. Speaker: Nicola Pagani (Liverpool)
Title: A wall-crossing formula for universal Brill-Noether classes
Abstract: We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (=over the moduli space of stable curves) Brill-Noether classes. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. The calculation involves constructing a resolution by means of subsequent blow-ups. If time permits, we will discuss the significance of our formula and potential applications. This is joint with Alex Abreu.
Wednesday 14 June 2023, 3pm. Speaker: Christian Boehning (Warwick)
Title: Equivariant birational geometry of cubic fourfolds and derived categories
Abstract: In the talk I will discuss several aspects of equivariant birationality from the perspective of derived categories. I will exhibit examples of nonlinearisable but stably linearisable actions of finite groups on smooth cubic fourfolds that show that the natural equivariant analogs of existing rationality conjectures for cubic fourfolds do not hold. Time permitting, I will also discuss some results about G-Del Pezzo surfaces and some classes of higher-dimensional G-Fano varieties.
This is joint work with Hans-Christian von Bothmer, Brendan Hassett and Yuri Tschinkel.
Wednesday 21 June 2023, 3pm. Speaker: Arend Bayer (Edinburgh)
Title: Non-commutative abelian surfaces and generalised Kummer varieties
Abstract: Polarised abelian surfaces vary in three-dimensional families. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations; moreover, based on general principles, one should expect to get "algebraic families" of their categories over four-dimensional bases.
Generalised Kummer varieties (GKV) are Hyperkaehler varieties arising from moduli spaces of stable sheaves on abelian surfaces. Polarised GKVs have four-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarities.
I present a construction that addresses both issues. We construct four-dimensional families of categories that are deformations of D^b(A) over an algebraic space. Moreover, each category admits a Bridgeland stability conditions, and from the associated moduli spaces of stable objects one can obtain every general polarised GKV, for every possible polarisation type of GKVs. Our categories are obtained from Z/2-actions on derived categories of K3 surfaces.
This is based on joint work with Laura Pertusi, Alex Perry and Xiaolei Zhao.
Wednesday 28 June 2023, 3pm. Speaker: Michael Wemyss (Glasgow)
Title: Stability conditions and the Crepant Transformation Conjecture via enhancing the movable cone
Abstract: In the first part of my talk, which may perplex you as to why its in an algebraic geometry seminar, I will explain how to construct a finite and infinite hyperplane arrangement from the data of a subset of nodes in an ADE Dynkin diagram. Sounds easy. It is. I'll then explain what this has to do with 3-fold flops. The infinite arrangement turns out to control (a) stability conditions, and (b) give a conceptual and visually pleasing proof of the Crepant Transformation Conjecture in this context. Parts are joint with Iyama, Hirano and Nabijou.