Algebraic Geometry Seminar

The algebraic geometry seminar in Term 1 2026/2027 will usually meet on Wednesdays at 3 pm, though we may sometimes change to allow speakers from other time zones.

All talks will be in B3.02.

A link to talks from the previous term is here.

The seminar is currently managed by Marco Fava. Please get in touch if you wish to suggest a speaker!

Wednesday 7 Oct 2026, 3pm. Speaker: Paul Johnson (Sheffield)

Title: Betti numbers of Hilbert schemes of points on orbifold surfaces.

Abstract: Building on Ellingrud and Strømme's calculation of the Betti numbers for the Hilbert scheme of points in the plane, in his master's thesis Göttsche gave product formulas for the Betti numbers of Hilbert schemes on any smooth surface. Since a smooth DM stack should be as well behaved as a smooth surface, one should expect similar product formulas for orbifold surfaces. Indeed, when G in SL_2, the components of Hilb_n([C^2/G]) are Nakajima quiver varieties, and their cohomologies assemble into representations of quantum groups. Surprisingly, not much work had been done for more general G. In joint work with Jørgen Vold Rennemo we remedy this, giving product formulas when G is abelian and some results beyond that.

In the first half (and then some) we review Ellingsrud and Strømme's calculation of the Betti numbers for the Hilbert scheme of points in the plane, explaining how this allows you to calculate the Betti numbers of Hilb [C^2/G] when G is abelian in terms of partitions, which allowed Gusein-Zade, Luengo, and Melle-Hernández to conjecture the first product formulas and for me to extend these formulas to all abelian groups. In the remaining time we sketch Jørgen's clever trick to prove them using the Crepant Resolution Conjecture. If time allows we'll touch on what we can see about the non-abelian case.

Wednesday 14 Oct 2026, 3pm. Speaker: none (Staff meeting)

Wednesday 21 Oct 2026, 3pm. Speaker: Veronica Arena (Cambridge)

Title: Logarithmic Hilbert schemes of points on a Curve

Abstract: Weighted blow-ups are a birational transformation that naturally appears in moduli spaces. One instance where this happens, is when studying the logarithmic Hilbert scheme of points on a curve $C$ equipped with a log structure. Today we will give a quick introduction to both weighted blow-ups and logarithmic Hilbert schemes of points on curves. Then we will focus our attention on the examples of two and three points on $(P^1|0)$ and will describe the wall crossings between the classical Hilbert scheme of points and the logarithmic ones via weighted blow-ups.

Wednesday 28 Oct 2026, 3pm. Speaker: Marco Fava (Warwick)

Title: The cohomology of fine compactified universal Jacobians

Abstract: The Jacobian of a smooth curve is an abelian variety parametrising its line bundles of degree 0, and classically, together with the theta divisor, determines the curve itself. If we drop the smoothness hypothesis, however, the Jacobian variety is, in general, not proper and many compactifications depending on stability conditions have been constructed over the last 50 years. In particular, modular compactifications of the universal Jacobian over the moduli spaces of stable genus g curves with n markings, have been recently classified for any g and n.

In this talk, we use this classification, to give a computational method for the cohomology of fine compactified universal Jacobians. We do so by describing it via the concurrent stratifications of the boundaries of the moduli space of stable curves and of the compactified Jacobian of a single curve. Then, for each stratum, we compute the Hodge-Euler characteristic that encodes its mixed Hodge structure and recover the cohomology of the full space by orbifold-smoothness and properness. We display the results obtained in the first genus 2 cases and announce a general formula for any genus and number of markings (work in progress with Siddarth Kannan and Terry Song).

Wednesday 4 Nov 2026, 3pm. Speaker: Nikon Kurnosov (UCL)

Title: Class VII surfaces with the help of foliations

Abstract: Complex surfaces are divided into the classes by their invariants, I will briefly recall this classification and focus on class VII surfaces, which are surfaces with Kodaira dimensions -\inf and b_1=1. Bogomolov theorem says that the ones with b_2=0 are only Hopf surfaces and Inoue. One of the essential tools to study them is to figure out what foliations look like. This happens to be the case as well for class VII surfaces with b_2>0. I will review technique of Brunella, of Spicer and myself and some other recent results.

Wednesday 11 Nov 2026, 3pm. Speaker: TBA

Wednesday 18 Nov 2026, 3pm. Speaker: Matthew Habermann (Imperial)

Wednesday 25 Nov 2026, 3pm. Speaker: Pierrick Bousseau (Oxford)

Wednesday 2 Dec 2026, 3pm. Speaker: Alexia Corradini (Cambridge)

Wednesday 9 Dec 2026, 3pm. Speaker: Simon Cooper (Imperial)