# Virtual Resolutions for a Product of Projective Spaces

## TL;DR

This paper develops shorter free complexes to better encode the geometry of subvarieties in products of projective spaces and toric varieties, addressing the complexity of traditional minimal free resolutions.

## Contribution

It introduces novel, shorter free complexes that improve the encoding of geometric properties over Cox rings in complex ambient spaces.

## Key findings

- Constructed shorter free complexes for products of projective spaces.
- Reduced complexity of resolutions compared to traditional minimal free resolutions.
- Enhanced understanding of geometric properties via these new complexes.

## Abstract

Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.