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.

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)

Wednesday 28 Oct 2026, 3pm. Speaker: Fatemeh Rezaee (Cambridge)

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

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: TBA