Multigraded Castelnuovo-Mumford Regularity

Details Multigraded Castelnuovo-Mumford Regularity Multigraded Castelnuovo-Mumford Regularity

2003-05-15

arxiv.org/abs/math/0305214v1

Journal fur die reine und angewandte Mathematik (Crelle's Journal) 571 (2004) 179-212

Mathematics

Commutative Algebra

30 pages, 5 figures

Scientific paper

10.1515/crll.2004.040

We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the associated subvariety. We also provide a criterion for testing if an ample line bundle on X gives a projectively normal embedding.

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