We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.

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...

We consider the set of forms of a toric variety over an arbitrary field: those varieties which become isomorphic to a toric variety after base field extension. In contrast to most previous work, we also consider arbitrary isomorphisms rather than just those that respect a torus action. We define an injective map from the set of forms of a toric variety to a non-abelian second cohomology set, which generalizes the usual Brauer class of a Severi-Brauer variety. Additionally, we define a map from the set of forms of a toric variety to the set of forms of a separable algebra along similar lines to a construction of A. Merkurjev and I. Panin. This generalizes both a result of M. Blunk for del Pezzo surfaces of degree 6, and the standard bijection between Severi-Brauer varieties and central simple algebras.

Abstract. We show that Y. Prokhorov's "Simple Finite Subgroups of the Cremona Group of Rank 3" implies that, over any field of characteristic 0, the essential dimensions of the alternating group, A 7 , and the symmetric group, S 7 , are 4.

We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0., the simple group of order 168, 7. S 5 , the symmetric group on 5 letters.Furthermore, any finite subgroup of these groups has essential dimension ≤ 2.A few remarks are in order. Remark 1.1. We do not classify all versal minimal rational G-surfaces; we only determine which groups appear. Different G-surfaces with the same group G may not be equivariantly birationally equivalent. There exist two versal S 5surfaces that are not equivariantly birationally equivalent: the Clebsch diagonal cubic (by a result of Hermite, see [14], [47] and [32]) and the del Pezzo surface of degree 5 (see the proof of Theorem 4.5). Other examples of this phenomenon can be found for abelian groups [46], and for versal actions of S 4 and A 5 [2].Remark 1.2. Essential dimension can be defined over any field. Dolgachev and Iskovskikh's classification, and many of our other references, take the base field to be C. We shall see in Lemma 2.1 below that this is sufficient to handle any algebraically closed field of characteristic 0.Remark 1.3. For algebraically closed fields of non-zero characteristic, the Enriques-Manin-Iskovskikh classification still holds. However, the Dolgachev-Iskovskikh classification no longer applies. Furthermore, unirational surfaces are not necessarily rational in this case, so the classification may be inadequate. See [51] for related discussion.