Syzygies of Curves in Products of Projective Spaces

TL;DR

This paper extends syzygy analysis of curves from projective spaces to products of projective spaces, establishing bounds on regularity based on degree and ambient dimension, with new links between multigraded resolutions and regularity.

Contribution

It introduces a framework for studying syzygies in products of projective spaces and proves an analogue of a key regularity bound for singular curves.

Findings

Bound on regularity of curves in products of projective spaces.

New results linking multigraded resolutions to regularity regions.

Extension of classical syzygy results to more complex ambient spaces.

Abstract

Motivated by toric geometry, we lift machinery for understanding syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space. To do so, we show new results linking the shape of multigraded resolutions of a sheaf to its regularity region.