Nef cones of Hilbert schemes of points on surfaces

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.