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

Theorems & Definitions (39)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3: See \ref{['thm:virtual-of-pair-stronger']} and \ref{['sec:cases-2-to-11']}
• Definition 1.4
• Proposition 1.5
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5