Algebraic Geometry Seminar

The algebraic geometry seminar in Term 2 2025/2026 will usually meet on Wednesdays at 3 pm, though we may sometimes change to allow speakers from other time zones.

Most talks will be either in B3.02 or MB0.07 -- see specific information below.

A link to talks from the previous term is here.

Wednesday 21 Jan 2026, 3pm. Speaker: Alexey Elagin (Sheffield) / *cancelled*

Title: Atomic semi-orthogonal decompositions for derived categories of surfaces

Abstract:

There is an old expectation that birational geometry (in particular, the Minimal Model Program) can and should be translated into the language of derived categories. For example, the blow-up at a smooth centre corresponds to adding several copies of the derived category of the centre to the derived category of the base. It is tempting then to extract a birational invariant from derived categories of algebraic varieties, however, there is an obstacle: semi-orthogonal decompositions of these categories are essentially non-unique (i.e., the Jordan-Hoelder property fails).

I will talk about a possible way to overcome this obstacle by constructing semi-orthogonal decompositions that we call ``atomic'' and that have two properties: they are (i) unique in a certain sense and (ii) compatible with birational morphisms (so can be applied to birationality questions). The theory is now established in dimension 2, over any perfect field and in G-equivariant setting. Its construction is based on the G-Minimal Model Program for surfaces and on the Sarkisov link decomposition. Time permitting, I will explain the role that ``atoms'' can play in the classification of geometrically rational surfaces over non-closed fields.

This is a joint work with Evgeny Shinder and Julia Schneider, see arXiv:2512.05064.

Wednesday 28 Jan 2026, 3pm. Speaker: Thomas Eckl (Liverpool)

Title: The complex geometry of crystallographic groups

Abstract:

Crystallographic groups describe the symmetries of patterns in the Euclidean space $\mathbb{E}^n$ that are periodic in n linearly independent directions, and thus appeared in myriads of decorations through the millennia, but also play a key role in studying physical and chemical properties of crystals.

In this talk, we first show that the classification of n-dimensional crystallographic groups can be reduced to the classification of n-dimensional orbifold K\"ahler klt pairs with vanishing first and second orbifold Chern class. In particular, such pairs can be described as quotients of complex tori by a finite complex linear group, and we present criteria that such groups must satisfy. Finally, we apply all this to reproduce the classical classification of wallpaper groups.

Wednesday 4 Feb 2026, 3pm. Speaker: Luca Kollmer (Warwick)

Title: Orbifolds by S_n-symmetric groups and the knockout game

Abstract: Let A be a finite diagonal Abelian subgroup of SL(C, d), and consider the quotient singularity (orbifold) C^d/A. Everyone knows the resolution of Du Val singularities when d = 2. Nakamura in 2001 introduced the A-Hilbert scheme, the moduli space of A-clusters, and showed that it is a smooth crepant resolution of C^3/A. Craw and Reid described in 2002 the knockout game for drawing the toric fan of A-Hilbert scheme. This game generalises easily to higher dimensions under the ‘all-even’ condition when the group is S_{d-2}-symmetric in the first d-2 coordinates. However, now the A-Hilbert scheme is at worst a blowup at ‘traps’ of the crepant resolution.

Wednesday 11 Feb 2026, 3pm. Speaker: Christian Boehning (Warwick)

Title: Atoms for the practical person

Abstract:

I will present a relatively elementary approach to (part of) the new theory of atoms due to Cavenaghi-Iritani - Kontsevich-Katzarkov-Pantev-Yu “for the practical person” and illustrate it with a few computations, e.g. for special cubic fourfolds. I will also discuss some new applications to non-linearisability and stable non-linearisability of some classes of G-varieties, for example G-del Pezzo surfaces. Time permitting I will mention some steps towards a possible elementary proof of (the consequences of) Iritani’s theorem needed for the applications. This is joint work in progress with Hans-Christian von Bothmer, Yuri Tschinkel, Zhijia Zhang.

Wednesday 18 Feb 2026, 4pm, room D1.07. Alexey Elagin (Sheffield)

Title: Atoms and birational geometry of surfaces over non-closed fields

Abstract: I will review old good birational classification of surfaces over non-closed fields using new categorical invariants called atoms. Atoms of a surface X are certain triangulated categories naturally associated with X, they appear as components of a semi-orthogonal decomposition of the derived category D^b(X). In most cases atoms of X determine X up to a birational isomorphism and in many cases - up to a biregular isomorphism. Most results in this direction were known by classics, but some are new: for example, we show that two birational minimal del Pezzo surfaces of degree 4 are isomorphic.
This is a joint work with Julia Schneider and Evgeny Shinder.

Wednesday 25 Feb 2026, 3pm. Speaker: Thibault Poiret (St Andrews)

Title: TBA

Abstract: TBA

Wednesday 4 Mar 2026, 3pm. Speaker: Hamid Abban (Nottingham)

Title: TBA

Abstract: TBA

Wednesday 11 Mar 2026, 3pm. Speaker: Parth Shimpi (Glasgow)

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Wednesday 18 Mar 2026, 3pm. Speaker: Hanfei Guo (Fudan University)

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