  Let X be a smooth projective surface of irregularity 0. The Hilbert scheme of
n points on X parameterizes zero-dimensional subschemes of X of length n. In
this paper, we discuss general methods for studying the cone of ample divisors
on the Hilbert scheme. We then use these techniques to compute the cone of
ample divisors on the Hilbert scheme 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 Macri.
