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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

References

[AG62]Andreotti, A. and Grauert, H., Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193–259; MR 0150342(27#343).CrossRefGoogle Scholar
[Bot59]Bott, R., On a theorem of Lefschetz, Michigan Math. J. 6 (1959), 211–216; MR 0215323(35#6164).CrossRefGoogle Scholar
[DPS96]Demailly, J.-P., Peternell, T. and Schneider, M., Holomorphic line bundles with partially vanishing cohomology, in Proceedings of the Hirzebruch 65 conference on algebraic geometry (Ramat Gan, 1993), Israel Mathematical Conference Proceedings, vol. 9 (Bar-Ilan University, 1996), 165–198; MR 1360502(96k:14016).Google Scholar
[Ful93]Fulton, W., Introduction to toric varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, NJ, 1993), The William H. Roever Lectures in Geometry;MR 1234037(94g:14028).CrossRefGoogle Scholar
[Har77]Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, vol. 52 (Springer, New York, 1977); MR 0463157(57#3116).CrossRefGoogle Scholar
[K{ü}r10]Küronya, A., Positivity on subvarieties and vanishing of higher cohomology, Preprint (2010), arXiv:1012.1102v1.Google Scholar
[Laz04a]Lazarsfeld, R., Positivity in algebraic geometry, I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48 (Springer, Berlin, 2004), Classical setting: line bundles and linear series; MR 2095471(2005k:14001a).Google Scholar
[Laz04b]Lazarsfeld, R., Positivity in algebraic geometry, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49 (Springer, Berlin, 2004), Positivity for vector bundles, and multiplier ideals; MR 2095472(2005k:14001b).Google Scholar
[Mat11]Matsumura, S., Asymptotic cohomology vanishing and a converse to the Andreotti–Grauert theorem on a surface, Preprint (2011), arXiv:1104.5313v1.Google Scholar
[Nak00]Nakamaye, M., Stable base loci of linear series, Math. Ann. 318 (2000), 837–847; MR 1802513(2002a:14008).CrossRefGoogle Scholar
[Ott11]Ottem, J. C., Ample subvarieties and q-ample divisors, Preprint (2011), arXiv:1105.2500v2.CrossRefGoogle Scholar
[Tot10]Totaro, B., Line bundles with partially vanishing cohomology, Preprint (2010), arXiv:1007.3955v1.Google Scholar