Versality of algebraic group actions and rational points on twisted varieties

Abstract

We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"><mml:semantics><mml:mi>X</mml:mi><mml:annotation encoding="application/x-tex">X</mml:annotation></mml:semantics></mml:math></inline-formula>(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<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>-versality for a prime integer<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"><mml:semantics><mml:mi>p</mml:mi><mml:annotation encoding="application/x-tex">p</mml:annotation></mml:semantics></mml:math></inline-formula>. An appendix, containing a letter from J.-P. Serre, puts the notion of versality in a historical perspective.