𝐹-split and 𝐹-regular varieties with a diagonalizable group action

Achinger, Piotr; Ilten, Nathan; Süss, Hendrik

doi:10.1090/jag/686

$F$-split and $F$-regular varieties with a diagonalizable group action

Authors: Piotr Achinger, Nathan Ilten and Hendrik Süss
Journal: J. Algebraic Geom. 26 (2017), 603-654
DOI: https://doi.org/10.1090/jag/686
Published electronically: December 9, 2016
MathSciNet review: 3683422
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Abstract | References | Additional Information

Abstract: Let $H$ be a diagonalizable group over an algebraically closed field $k$ of positive characteristic, and $X$ a normal $k$-variety with an $H$-action. Under a mild hypothesis, e.g. $H$ a torus or $X$ quasiprojective, we construct a certain quotient log pair $(Y,\Delta )$ and show that $X$ is $F$-split ($F$-regular) if and only if the pair $(Y,\Delta )$ is $F$-split ($F$-regular). We relate splittings of $X$ compatible with $H$-invariant subvarieties to compatible splittings of $(Y,\Delta )$, as well as discussing diagonal splittings of $X$. We apply this machinery to analyze the $F$-splitting and $F$-regularity of complexity-one $T$-varieties and toric vector bundles, among other examples.

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Additional Information

Piotr Achinger
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: piotr.achinger@gmail.com

Nathan Ilten
Affiliation: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, British Columbia V5A 1S6, Canada
MR Author ID: 864815
Email: nilten@sfu.ca

Hendrik Süss
Affiliation: School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
MR Author ID: 834968
Email: hendrik.suess@manchester.ac.uk

Received by editor(s): April 22, 2015
Received by editor(s) in revised form: November 30, 2015, and December 10, 2015
Published electronically: December 9, 2016
Article copyright: © Copyright 2016 University Press, Inc.