Versality of algebraic group actions and rational points on twisted varieties

Details Versality of algebraic group actions and rational points on twisted varieties Versality of algebraic group actions and rational points on twisted varieties

2011-09-28

arxiv.org/abs/1109.6093v1

Mathematics

Algebraic Geometry

31 pages. With an appendix by J.-P. Serre

Scientific paper

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.

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