  Let $X$ be the projective plane, a Hirzebruch surface, or a general $K3$
surface. In this paper, we study the birational geometry of various nested
Hilbert schemes of points parameterizing pairs of zero-dimensional subschemes
on $X$. We calculate the nef cone for two types of nested Hilbert schemes:
$X^{[n+1,n]}$ and universal families. As an application, we recover a theorem
of Butler on syzygies on Hirzebruch surfaces.
