Nef cones of Hilbert schemes of points on surfaces

TL;DR

This paper develops methods to compute nef cones of Hilbert schemes of points on surfaces, applying them to new classes of surfaces using Bridgeland stability and positivity techniques.

Contribution

It introduces general techniques for determining nef cones and explicitly computes them for several surfaces where the cones were previously unknown.

Findings

Computed nef cones for Hilbert schemes on various surfaces.

Extended understanding of ample divisors on Hilbert schemes.

Applied Bridgeland stability to nef cone calculations.

Abstract

Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of n points on X parameterizes zero-dimensional subschemes of X of length n. In this paper, we discuss general methods for studying the cone of ample divisors on the Hilbert scheme. We then use these techniques to compute the cone of ample divisors on the Hilbert scheme for several surfaces where the cone was previously unknown. Our examples include families of surfaces of general type and del Pezzo surfaces of degree 1. The methods rely on Bridgeland stability and the Positivity Lemma of Bayer and Macri.