On virtual resolutions of points in a product of projective spaces

Abstract

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 ${\mathbb{P}}^n\times {\mathbb{P}}^m$.

18 pages