Nef cones of Hilbert schemes of points on surfaces

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.

Corrected an error in section 5: the action of the Weyl group on the extremal rays of Nef(X) is not transitive, and instead has two orbits (see Proposition 5.2). All the results from the section still hold after small modifications to the proofs