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m-Blocks Collections and Castelnuovo-mumford Regularity in multiprojective spaces

Published online by Cambridge University Press:  11 January 2016

L. Costa  and
R. M. Miró-Roig
L. Costa
Affiliation:
Facultat de Matemàtiques Departament d’Algebra i Geometria Gran Via de les CortsCatalanes 585 08007Barcelona Spaincosta@ub.edu
R. M. Miró-Roig
Affiliation:
Facultat de Matemàtiques Departament d’Algebra i Geometria Gran Via de les CortsCatalanes 585 08007Barcelona Spainmiro@ub.edu
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Abstract

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The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the n-block collection B. 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 compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on ℙn and for the n-block collection Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space ℙn1x…x ℙnr with respect to a suitable n1 +…+ nr-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].

Keywords

14F05
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 186 , 2007 , pp. 119 - 155
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2007

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