Versality of algebraic group actions and rational points on twisted varieties

By Alexander Duncan, Zinovy Reichstein. 2011

55 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/78f3075d9f77289bfe9d9ff34752bc4319d7268e

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In Geometry and Topology

1. Alexander S. Merkurjev
2. N. Karpenko
3. B. Totaro

In Mathematical Physics

1. Alexander S. Merkurjev
2. N. Karpenko

Citations in the corpus

• Essential dimension: a survey (2013, methodology): “Following [20], we call a G-scheme X versal if for every generically free G-scheme X ′ with the field F (X ′)G infinite and every dense open G-invariant set U ⊂ X, there is a G-equivariant rational morphism X ′ 99K U .”