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Big q-ample line bundles

Part of: Cycles and subschemes

Published online by Cambridge University Press:  19 March 2012

Morgan V. Brown
Morgan V. Brown*
Affiliation:
Department of Mathematics, University of California, Berkeley, 970 Evans Hall, Berkeley, CA 94720-3840, USA (email: [email protected])
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Abstract

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A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf ℱ on X, there exists an integer m0 such that m≥m0 implies Hi (X,ℱ⊗𝒪(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.

Keywords

augmented base locipartial amplitudelinear series

MSC classification

Secondary: 14C20: Divisors, linear systems, invertible sheaves
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 3 , May 2012 , pp. 790 - 798
Copyright
Copyright © Foundation Compositio Mathematica 2012

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