Versality of algebraic group actions and rational points on twisted varieties

TL;DR

This paper explores the concept of versality in algebraic group actions, establishing equivalences with rational points on twisted varieties, and applies these insights to understand rational points and p-versality.

Contribution

It formalizes various notions of versality and proves their equivalence to properties of rational points on twisted forms of varieties, including p-versality, with historical context from Serre.

Findings

Versality notions are equivalent to rational point properties on twisted varieties.

Results apply to both general and p-versality cases.

Historical perspective provided by Serre's letter.

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.