  For finite sets of points in $\mathbb{P}^n \times \mathbb{P}^m$, we produce
short virtual resolutions, as introduced by Berkesch--Erman--Smith. We first
intersect with a sufficiently high power of one set of variables for points in
$\mathbb{P}^n \times \mathbb{P}^m$ to produce a virtual resolution of length
$n+m$. Then, we describe an explicit virtual resolution of length 3 for a set
of points in sufficiently general position in $\mathbb{P}^1 \times
\mathbb{P}^2$, via a subcomplex of a free resolution. This first result
generalizes to $\mathbb{P}^n \times \mathbb{P}^m$ work of Harada--Nowroozi--Van
Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl
and Booms-Peot, which were both for $\mathbb{P}^1 \times \mathbb{P}^1$. Along
the way, we also note an explicit relationship between Betti numbers and higher
difference matrices of bigraded Hilbert functions for $\mathbb{P}^n \times
\mathbb{P}^m$.
