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 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. 2.4. It is vital that algebraic groups are linear throughout this paper. For example, if A = {1} is an abelian variety, then no versal A-torsor can exist. Otherwise the exponent of every element of H 1 (K, A), for any field K/k, would divide the exponent of the versal torsor. However, there are A-torsors of arbitrarily high exponent (over suitable K); cf. [Bro07, Section 3].Proposition 2.5. If X is a versal irreducible G-variety, then X is geometrically irreducible.Proof. It suffices to show that X k s is irreducible, where k s denotes a separable closure of k; see [Har77, Exercise 2.3.15(a)]. Let X 1 , . . . , X n denote the irreducible components of X k s . We want to show that n = 1. We will argue by contradiction. Assume n 2. Since X is irreducible over k, the absolute Galois group Gal(k) permutes X 1 , . . . , X n transitively. Thus Y := X 1 ∩ · · · ∩ X n is a closed G-invariant k-subvariety of X (possibly empty) and X = Y .Let V be a generically free linear G-representation with k(V ) G infinite. By Remark 2.1 there exists a G-equivariant rational k-map f : V X \ Y . Since V is geometrically irreducible, the image of f is contained in one of the components X i . Since Gal(k) transitively permutes the components, the image of f is also contained in X 2 , . . . , X n and thus in Y , a contradiction.Remark 2.6. Suppose X is of minimal dimension among generically free versal G-varieties. Then X must be very versal. To see this, let V be a generically free linear representation of G, as in Remark 2.3. Let U ⊂ X be a G-invariant open subvariety which is the total space of a G-torsor U → B. Then U is weakly versal and, by Remark 2.1, there exists a G-equivariant rational map f : V U . The closure Z of the image of f is very versal and, since U is a G-torsor, the action on Z is generically free. By minimality of dim(X), we conclude that dim(Z) = dim(X) and thus f is dominant.The minimal value of dim(X)−dim(G), as X ranges over the very versal (or equivalently, versal) generically free G-varieties, is called the essential dimension of G and is denoted by ed(G); see [BF03], [Ser03, Section 5] or [Rei10].Remark 2.7. The following criterion gives a convenient way (often the only known way) to show that a given action is not versal.Let X be a projective irreducible weakly versal G-variety defined over k. Suppose H ⊂ G is a closed k-subgroup such that every finite-dimen...

Joubert's theorem fails in characteristic 2

“…Otherwise X would have a G-fixed point; see [DR,Remark 2.7]. Moreover, in view of Remark (3) above, the G-action on X is not even p-versal.…”

Pseudo-reflection groups and essential dimension

“…The G-orbit of y i is a 0-cycle of degree prime to p i . Thus for any given prime p, the degree of one of these orbits is prime to p. By Duncan and Reichstein [12, Corollary 8.6(b)], this implies that Y is p-versal for every p. Hence, pmed(G) dim(Y ) = d, and the proposition follows.…”

“…Case 1: Assume p = q. Recall that the G-action on X d is p-versal if and only if the G p -action on X d is p-versal, where G p is a Sylow p-subgroup of G; see [12,Corollary 8.6]. Since q > |G|, we have G q = {1}.…”

“…Case 2: Now assume p = q. To show that the G-action on X d is p-versal, it suffices to prove that for every field extension K/k, with K infinite, and every G-torsor T → Spec(K), the twisted K-variety T X d contains a 0-cycle Z, whose degree over K is a power of q (and thus prime to p); see [12,Section 8].…”

Essential dimension of inseparable field extensions

“…Here the subscript "ver" is meant to indicate that A ver /K ver is a versal object for Alg Λ = H 1 ( * , G). For a discussion of versal torsors, see Section I.5 in [Ser03] or [DR15].…”