#### Vol. 10, No. 4, 2016

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
Nef cones of Hilbert schemes of points on surfaces

### Barbara Bolognese, Jack Huizenga, Yinbang Lin, Eric Riedl, Benjamin Schmidt, Matthew Woolf and Xiaolei Zhao

Vol. 10 (2016), No. 4, 907–930
##### Abstract

Let $X$ be a smooth projective surface of irregularity $0$. The Hilbert scheme ${X}^{\left[n\right]}$ of $n$ points on $X$ parametrizes zero-dimensional subschemes of $X$ of length $n$. We discuss general methods for studying the cone of ample divisors on ${X}^{\left[n\right]}$. We then use these techniques to compute the cone of ample divisors on ${X}^{\left[n\right]}$ 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 Macrì.

A correction was posted on 16 June 2018 in an online supplement.
##### Keywords
Hilbert schemes, surfaces, nef cone, ample cone, birational geometry, Bridgeland stability
##### Mathematical Subject Classification 2010
Primary: 14C05
Secondary: 14E30, 14J29, 14J60
##### Supplementary material

Errata for this article

##### Milestones
Received: 1 November 2015
Revised: 12 March 2016
Accepted: 7 May 2016
Published: 20 June 2016
##### Authors
 Barbara Bolognese Department of Mathematics Northeastern University Boston, MA 02115 United States Jack Huizenga Department of Mathematics The Pennsylvania State University University Park, PA 16802 United States Yinbang Lin Department of Mathematics Northeastern University Boston, MA 02115 United States Eric Riedl Department of Mathematics, Statistics, and CS University of Illinois Chicago, IL 60607 United States Benjamin Schmidt Department of Mathematics The Ohio State University Columbus, OH 43210 United States Matthew Woolf Department of Mathematics, Statistics, and CS University of Illinois Chicago, IL 60607 United States Xiaolei Zhao Department of Mathematics Northeastern University Boston, MA 02115 United States