On the birational geometry of Hilbert schemes of points and Severi divisors

TL;DR

This paper investigates the birational geometry of Hilbert schemes of points on surfaces, focusing on the stable base locus decomposition and Severi divisors, and examines the weak Lefschetz principle in this context.

Contribution

It demonstrates that the restriction of augmented stable base loci aligns with the stable base locus decomposition of subvarieties, and computes classes of Severi divisors affecting the decomposition.

Findings

Restriction of augmented stable base loci matches the stable base locus decomposition.

Severi divisors induce walls in the stable base locus decomposition.

Results apply to Hilbert schemes on the projective plane.

Abstract

We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if the weak Lefschetz Principle holds. Even though the stable base loci in $N^1(X^{[n]})$ fails to provide information about how the two decompositions interact, we show that the restriction of the augmented stable base loci of $X^{[n]}$ to $Y^{[n]}$ is equal to the stable base locus decomposition of $Y^{[n]}$. We also exhibit effective divisors induced by Severi varieties. We compute the classes of such divisors and observe that in the case that $X$ is the projective plane, these divisors yield walls of the SBLD for some cases.