Finite groups of essential dimension 2

Abstract

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.

Versality of algebraic group actions and rational points on twisted varieties

“…Versal and closely related "generic" objects (cf. Remark 2.8) naturally arise in many parts of algebra and algebraic geometry, such as the theory of central simple algebras [Pro67,Ami72,Sal99], Galois theory [JLY02], and the study of algebraic surfaces [Dun09,Tok06]. For a historical perspective we refer the reader to the appendix.…”

“…Here one generator of Z/2Z × Z/2Z takes (x : y) ∈ P 1 to (−x : y) and the other to (y : x). Arguments of this type are used extensively, e.g., in [Dun09], [Dun10], [Bea11] and [Tok06].…”

Equivariant unirationality of del Pezzo surfaces of degree 3 and 4

“…All of these surfaces are toric varieties. By Corollary 3.6 of [Dun13], we see that (a) and (c) of Theorem 1.4 are equivalent (recall that G-versal and G-unirational are equivalent here by Proposition 2.1.) In particular, we may assume that G is a p-group.…”

“…The finite groups of essential dimension 2 were classified in Theorem 1.1 of [Dun13]. Thus, we know the groups G for which a G-unirational surface exists, but we do now know whether a given G-surface is G-unirational.…”

“…By Proposition 3.10 of [Dun13], any p-group acts on a smooth toric surface as a subgroup of (C × ) 2 ⋊ H where H is a finite subgroup of GL 2 (Z). Since any finite subgroup of GL 2 (Z) has order divisible by 2 or 3, if G is a pgroup where p ≥ 5 then G is a subgroup of (C × ) 2 .…”