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: Roots on curves and their combinatorics

Abstract: Several moduli spaces M related to curves or abelian varieties admit proper maps to the moduli of curves Mg,n or of principally polarized abelian varieties Ag, but are not proper themselves because Mg,n and Ag are not. It can be difficult to find compactifications of M which keep the good properties of M. I will discuss modern techniques to find such compactifications. These techniques give rise to natural coverings of the moduli space of stable curves, and I will discuss the combinatorial aspects of the geometry of these coverings.

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

Title: K-stability of Fano varieties, old and new

Abstract: K-stability has become a buzzword for those working with it and a mystery for those who don’t. To justify the former and to demystify the latter, I will give an introductory and survey-like talk on the topic of K-stability of Fano varieties illustrated with examples.

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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