#### 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 ${X}^{\left[n\right]}$ 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