On the universal family of Hilbert schemes of points on a surface

TL;DR

This paper investigates the geometric and singularity properties of the universal family over Hilbert schemes of points on a surface, revealing non-$Q$-Gorenstein rational singularities and bounds on Samuel multiplicity.

Contribution

It demonstrates that the universal family has non-$Q$-Gorenstein rational singularities and provides a formula for Samuel multiplicity in terms of the socle dimension.

Findings

Universal family has non-$Q$-Gorenstein rational singularities.

Samuel multiplicity at a point can be computed via socle dimension.

Samuel multiplicity is bounded above by the number of points, n.

Abstract

For a smooth quasi-projective surface $X$ and an integer $n\ge 3$, we show that the universal family $Z^n$ over the Hilbert scheme $\text{Hilb}^{n}(X)$ of $n$ points has non $\mathbb{Q}$-Gorenstein, rational singularities, and that the Samuel multiplicity $\mu$ at a closed point on $Z^n$ can be computed in terms of the dimension of the socle. We also show that $\mu\le n$.