H.B. Snodgras (Cambridge):
Cohomology of punctual Hilbert schemes of smooth projective surfaces
Time and place
Monday, 4.3.24, 10:15-11:15, Raum 404, Ernst-Zermelo-Str. 1
Abstract
The punctual Hilbert scheme X^[n] of a projective scheme X over a field k is a scheme parameterising the closed subschemes of X of length n; loosely speaking, those that have n points when counted with multiplicity. It turns out that if X is smooth, X^[n] is necessarily smooth as well if and only if dim X < 3, making the case where X is a smooth projective surface of particular interest. Remarkable work by L. Göttsche demonstrated that if k is C or the algebraic closure of a finite field, the Betti numbers and the Euler characteristic of X^[n] can be written in terms of explicit generating functions related to modular forms. This talk will review some properties of punctual Hilbert schemes in general, study the special case of projective surfaces and discuss their cohomology.