# Virtual Complete Intersections in $\mathbb{P}^1 \times \mathbb{P}^1$

## TL;DR

This paper explores conditions under which sets of points in the product of two projective lines have virtual resolutions that are Koszul complexes on regular sequences, extending understanding of algebraic structures in this setting.

## Contribution

It identifies specific conditions that determine when sets of points in $P^1 imes P^1$ have virtual resolutions as Koszul complexes on regular sequences.

## Key findings

- Certain sets of points guarantee a Koszul complex virtual resolution.
- Other sets of points are proven not to have such a resolution.
- Provides a characterization of points with these properties.

## Abstract

The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.