Regularity of smooth curves in biprojective spaces

Maclagan and Smith [D. Maclagan, G.G. Smith, Multigraded Castelnuovo–Mumford regularity, J. Reine Angew. Math. 571 (2004) 179–212] developed a multigraded version of Castelnuovo–Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve C⊆Pa×Pb (a,b⩾2) of bidegree (d1,d2) with nondegenerate birational projections the ideal sheaf IC|Pa×Pb is (d2−b+1,d1−a+1)-regular. We also give an example showing that in some cases this bound is the best possible.