The Birational Geometry of the Hilbert Scheme of Points on Surfaces

Abstract

In this paper, we study the birational geometry of the Hilbert scheme of points on a smooth, projective surface, with special emphasis on rational surfaces such as \({\mathbb{P}}^{2}, {\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) and \(\mathbb{F}_{1}\). We discuss constructions of ample divisors and determine the ample cone for Hirzebruch surfaces and del Pezzo surfaces with K 2≥2. As a corollary, we show that the Hilbert scheme of points on a Fano surface is a Mori dream space. We then discuss effective divisors on Hilbert schemes of points on surfaces and determine the stable base locus decomposition completely in a number of examples. Finally, we interpret certain birational models as moduli spaces of Bridgeland-stable objects. When the surface is \({\mathbb{P}}^{1} \times {\mathbb{P}}^{1}\) or \(\mathbb{F}_{1}\), we find a precise correspondence between the Mori walls and the Bridgeland walls, extending the results of Arcara et al. (The birational geometry of the Hilbert scheme of points on \({\mathbb{P}}^{2}\) and Bridgeland stability, arxiv:1203.0316, 2012) to these surfaces.