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. © 2009 Elsevier Inc. All rights reserved.
Regularity of smooth curves in biprojective spaces
Lozovanu V.
2009-01-01
Abstract
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. © 2009 Elsevier Inc. All rights reserved.
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