Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

$ 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
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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
Email: nilten@sfu.ca

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

DOI: https://doi.org/10.1090/jag/686
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.

American Mathematical Society