Nef cones of nested Hilbert schemes of points on surfaces

TL;DR

This paper investigates the birational geometry of nested Hilbert schemes on surfaces like the projective plane, Hirzebruch, and K3 surfaces, calculating nef cones and applying results to syzygies.

Contribution

It computes nef cones for specific nested Hilbert schemes and applies these results to recover a theorem on syzygies for Hirzebruch surfaces.

Findings

Nef cones are explicitly calculated for $X^{[n+1,n]}$ and universal families.

The work recovers Butler’s theorem on syzygies on Hirzebruch surfaces.

Provides insights into the birational geometry of nested Hilbert schemes.

Abstract

Let $X$ be the projective plane, a Hirzebruch surface, or a general $K3$ surface. In this paper, we study the birational geometry of various nested Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes on $X$. We calculate the nef cone for two types of nested Hilbert schemes: $X^{[n+1,n]}$ and universal families. As an application, we recover a theorem of Butler on syzygies on Hirzebruch surfaces.