On virtual resolutions of points in a product of projective spaces

For finite sets of points in Pⁿ × Pᵐ, 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 Pⁿ × Pᵐ 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 P¹ × P², via a subcomplex of a free resolution. This first result generalizes to Pⁿ × Pᵐ work of Harada--Nowroozi--Van Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl and Booms-Peot, which were both for P¹ × P¹. Along the way, we also note an explicit relationship between Betti numbers and higher difference matrices of bigraded Hilbert functions for Pⁿ × Pᵐ.