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 $C\subseteq {\P }^a\times {\P }^b$ $(a,b\ge 2)$ 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.

11 pages