Regularity of smooth curves in biprojective spaces

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.