Georg Oberdieck (KTH Royal Institute of Technology)

Abstract: Let S be a smooth complex projective surface. The Hilbert schemes of points on S parametrizes the zero-dimensional closed subschemes of S. One way to describe it is as a crepant resolution of the symmetric product. In particular, it is smooth. The cohomology of the Hilbert scheme is well-understood as an irreducible representation of a Heisenberg algebra associated to the cohomology of the surface S. Much less is known about the quantum cohomology, which is a deformation of the classical cup product whose structure constants are (virtual) counts of rational curves with prescribed incidence conditions.

In the first part of the talk I will give a very basic introduction to the geometry of the Hilbert scheme and its cohomology. The main tool that we introduce are the Nakajima operators. In the main part, I will present recent work with Aaron Pixton in which we compute the quantum multiplication with a divisor on the Hilbert scheme of points on an elliptic surface. This is the first projective surface (with non-trivial Gromov-Witten theory), where the quantum multiplication with a divisor on the Hilbert scheme is described.