Relative Hilbert Scheme of Points

TL;DR

This paper computes the Betti numbers of the relative Hilbert scheme of points on a surface relative to a divisor, providing explicit generators and relations in specific cases like the projective plane.

Contribution

It offers the first explicit calculation of Betti numbers for relative Hilbert schemes of points and describes their cohomology structure in particular instances.

Findings

Betti numbers of the relative Hilbert scheme are computed.

Explicit generators and relations for cohomology are provided for b7b7b7

Results apply to the case of b7b7b7 and a line in b7b7b7.

Abstract

Let $D$ be a smooth divisor on a non singular surface $S$. We compute Betti numbers of the relative Hilbert scheme of points of $S$ relative to $D$. In the case of $\PP^2$ and a line in it, we give an explicit set of generators and relations for the cohomology groups of this space.