Vol. 10, No. 4, 2016

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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[n] 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[n]. We then use these techniques to compute the cone of ample divisors on X[n] 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.

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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