The algebraic geometry seminar in Term 1 2024/2025 will usually meet on Wednesdays at 3pm, though we may sometimes change to allow speakers from other time zones.

Each talk will be either in B3.02 or MB0.07 -- see specific information below.

A link to talks from the previous term is here.

Wednesday 23 October 2024, 3pm. Speaker: Will Donovan (Tsinghua University)

Title: Derived symmetries for crepant resolutions of hypersurfaces

Abstract: Given a singularity with a crepant resolution, a symmetry of the derived category of coherent sheaves on the resolution may often be constructed using the formalism of spherical functors. I will introduce this, and new work (arXiv:2409.19555) on general constructions of such symmetries for hypersurface singularities. This builds on previous results with Segal, and is inspired by work of Bodzenta-Bondal.

Wednesday 30 October 2024. Speaker: -

Title: -

Abstract: -

Wednesday 6 November 2024, 3pm. Speaker: Tarig Abdelgadir (Loughborough)

Title: The McKay correspondence via VGIT (case D4)

Abstract: For a Kleinian singularity, the McKay correspondence famously relates the orbifold cover of the singularity to a crepant resolution. In type A, both are toric and it is easy to write down a GIT problem which produces both the orbifold and the geometric resolution as possible quotients. However, no such construction seems to be known for types D and E. I'll describe how we fill this gap for the simplest non-trivial case D4. The construction is inspired by Tannaka duality and sets out a strategy to tackle general types D and E. This is joint work with Ed Segal.

Wednesday 13 November 2024. Speaker: Tiago Duarte Guerreiro (Paris-Saclay)

Title: On hypersurfaces in projective bundles

Abstract: Mori dream spaces are a special kind of varieties introduced by Hu and Keel in 2000 that enjoy very good properties with respect to the minimal model program. On the other hand, not many classes of examples of these are known. In this talk we introduce general hypersurfaces in certain projective bundles of Picard rank 2 and show that (some of) these are Mori dream spaces, partially generalising Ottem's result on hypersurfaces in products of projective spaces.

Wednesday 20 November 2024. Speaker: Alapan MukhopadyayLink opens in a new window (EPFL)

Title: Generators of bounded derived categories using the Frobenius map.

Abstract: Since the appearance of Bondal- van den Bergh’s work on the representability of functors, proving existence of strong generators of the bounded derived category of coherent sheaves on a scheme has been a central problem. While for a quasi-excellent, separated scheme the existence of strong generators is established, explicit examples of such generators are not common. In this talk, we show that explicit generators can be produced in prime characteristics using the Frobenius pushforward functor. As a consequence, for affine schemes,
we show that the pushforward of the structure sheaf via a high enough iterate of the Frobenius is a generator. This recovers Kunz’s characterization of regularity using Frobenius. We will discuss examples indicating that in contrast
to the affine situation, for a smooth projective scheme whether some Frobenius pushforward of the structure sheaf is a generator, depends on the geometry of the underlying scheme. Part of the talk is based on a joint work with Matthew
Ballard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.

(UNUSUAL TIME) Wednesday 27 November 2024, 1pm, MB0.07. Speaker: Elena Denisova (Edinburgh)

Title: Delta-invariants of Du Val del Pezzo surfaces

Abstract: It is known that a Fano variety with “mild” singularities admits a Kahler Einstein metric if and only if it is K-polystable. For two-dimensional Fano varieties (del Pezzo surfaces) Tian and Yau proved that a smooth del Pezzo surface is K-polystable if and only if it is not a blow up of P2 in one or two points. A lot of research was done for threefolds however, not everything is known and often the problem can be reduced to computing δ-invariant of (possibly singular) del Pezzo surfaces.

In my talk I will describe the status of the problem, present an example of computation of δ-invariant, show the example of application of this result for a singular Fano threefold and explain a possible direction for future research.

Wednesday 4 December 2024. Speaker: -

Title: -

Abstract: -