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Submission: 2011, Sep 30, revised: 2012, Jul 15
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, by J.-P. Serre, puts the notion of versality in a historical perspective.
2010 Mathematics Subject Classification: Primary: 14L30, 14G05. Secondary: 14M17, 14H10, 14M25
Keywords and Phrases: algebraic group, group action, torsor, versality, twisting, rational point, homogeneous space, universal torsor, toric variety, quadric, cubic hypersurface
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