Regularity of smooth curves in biprojective spaces

TL;DR

This paper investigates the regularity properties of smooth curves in biprojective spaces, establishing bounds on their ideal sheaves' regularity based on their bidegree and projections.

Contribution

It proves a new regularity bound for smooth curves in biprojective spaces using a multigraded Castelnuovo-Mumford regularity framework.

Findings

Established explicit regularity bounds for smooth curves in biprojective spaces.

Provided examples demonstrating the optimality of the bounds.

Extended the understanding of geometric properties of curves via multigraded regularity.

Abstract

Maclagan and Smith \cite{MaclaganSmith} developed a multigraded version of Castelnuovo-Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve $C\subseteq \P^a\times\P^b$ $(a, b\geq 2)$ of bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf $\mathcal{I}_{C|\P^a\times\P^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give an example showing that in some cases this bound is the best possible.