Virtual resolutions of points in $\mathbb{P}^1 \times \mathbb{P}^1$

TL;DR

This paper constructs explicit virtual resolutions for sets of points in the product of two projective lines, providing a simplified and effective method that depends only on the number of points, enhancing previous results.

Contribution

It describes a virtual resolution for general point sets in b^1 b^1 that depends solely on the number of points and improves existing bounds for such resolutions.

Findings

Provides a virtual resolution depending only on b^1 b^1 point count.

Offers an effective bound for virtual resolutions of length two.

Enhances previous existence results for point ideals in b^1 b^1.

Abstract

We explore explicit virtual resolutions, as introduced by Berkesch, Erman, and Smith, for ideals of sets of points in $\mathbb{P}^1 \times \mathbb{P}^1$. Specifically, we describe a virtual resolution for a sufficiently general set of points $X$ in $\mathbb{P}^1 \times \mathbb{P}^1$ that only depends on $|X|$. We also improve an existence result of Berkesch, Erman, and Smith in the special case of points in $\mathbb{P}^1 \times \mathbb{P}^1$; more precisely, we give an effective bound for their construction that gives a virtual resolution of length two for any set of points in $\mathbb{P}^1 \times \mathbb{P}^1$.