  We give an explicit formula for Euler characteristics of line bundles on the
Hilbert scheme of points on $\mathbb{P}^1\times\mathbb{P}^1$. Combined with
structural results of Ellingsrud, G\"ottsche, and Lehn, this determines the
Euler characteristic of any line bundle on the Hilbert scheme of points on any
smooth, projective surface. We also give an enumerative description of the
dimensions of spaces of global sections of ample line bundles on Hilbert
schemes of points on Hirzebruch surfaces, extending the polytope-line bundle
correspondence on the underlying toric surface.
