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THE GAMMA CONSTRUCTION AND ASYMPTOTIC INVARIANTS OF LINE BUNDLES OVER ARBITRARY FIELDS

Part of: (Co)homology theory General commutative ring theory Cycles and subschemes Local theory

Published online by Cambridge University Press:  23 September 2019

TAKUMI MURAYAMA Open the ORCID record for TAKUMI MURAYAMA [Opens in a new window]
TAKUMI MURAYAMA*
Affiliation:
Department of Mathematics, Princeton University, Princeton, NJ 08544-1000, USA email takumim@math.princeton.edu
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Abstract

We extend results on asymptotic invariants of line bundles on complex projective varieties to projective varieties over arbitrary fields. To do so over imperfect fields, we prove a scheme-theoretic version of the gamma construction of Hochster and Huneke to reduce to the setting where the ground field is $F$-finite. Our main result uses the gamma construction to extend the ampleness criterion of de Fernex, Küronya, and Lazarsfeld using asymptotic cohomological functions to projective varieties over arbitrary fields, which was previously known only for complex projective varieties. We also extend Nakayama’s description of the restricted base locus to klt or strongly $F$-regular varieties over arbitrary fields.

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 14F17: Vanishing theorems 14B05: Singularities
Type
Article
Information
Nagoya Mathematical Journal , Volume 242 , June 2021 , pp. 165 - 207
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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Footnotes

This material is based upon work supported by the National Science Foundation under grant nos. DMS-1501461 and DMS-1701622.

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