Versality of algebraic group actions and rational points on twisted varieties

Duncan, Alexander; Reichstein, Zinovy

doi:10.1090/S1056-3911-2015-00644-0

Authors: Alexander Duncan and Zinovy Reichstein
Journal: J. Algebraic Geom. 24 (2015), 499-530
DOI: https://doi.org/10.1090/S1056-3911-2015-00644-0
Published electronically: March 20, 2015
MathSciNet review: 3344763
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Abstract | References | Additional Information

Abstract: We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety $X$. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of $X$ (existence of rational points, existence of a dense set of rational points, etc.). We give applications of this equivalence in both directions to study versality of group actions and rational points on algebraic varieties. We obtain similar results on $p$-versality for a prime integer $p$. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.

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Additional Information

Alexander Duncan
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
MR Author ID: 854896
Email: arduncan@umich.edu

Zinovy Reichstein
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
MR Author ID: 268803
Email: reichst@math.ubc.ca

Received by editor(s): July 15, 2012
Received by editor(s) in revised form: April 29, 2013
Published electronically: March 20, 2015
Additional Notes: The first author was partially supported by National Science Foundation RTG grants DMS 0838697 and DMS 0943832. The second author was partially supported by National Sciences and Engineering Research Council of Canada Discovery grant 250217-2012.
Article copyright: © Copyright 2015 University Press, Inc.