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Geometric collections and Castelnuovo–Mumford regularity

Published online by Cambridge University Press:  01 November 2007

L. COSTA  and
R. M. MIRÓ–ROIG
L. COSTA*
Affiliation:
Facultat de Matemàtiques, Departament d'Algebra i Geometria, Gran Via de les Corts, Catalanes 585, 08007 Barcelona, Spain. email: [email protected], [email protected]
R. M. MIRÓ–ROIG*
Affiliation:
Facultat de Matemàtiques, Departament d'Algebra i Geometria, Gran Via de les Corts, Catalanes 585, 08007 Barcelona, Spain. email: [email protected], [email protected]
*
†Partially supported by MTM2004-00666.
‡Partially supported by MTM2004-00666.
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Abstract

The paper begins by overviewing the basic facts on geometric exceptional collections. Then we derive, for any coherent sheaf on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo–Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf on X with respect to σ. We show that the basic formal properties of the Castelnuovo–Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on and for a suitable geometric collection of coherent sheaves on both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo–Mumford regularity of their extension by zero in .

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 3 , November 2007 , pp. 557 - 578
Copyright
Copyright © Cambridge Philosophical Society 2007

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