Finite Simple Groups of Small Essential Dimension

Beauville, Arnaud

doi:10.1007/978-3-319-05254-0_16

Cited by 10 publications

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“…Since φ is irreducible, no point of X can be fixed by G. However, from [Bea11] we see that X has a G p -fixed point for any Sylow p-subgroup G p of G. Hence, by Corollary 10.6, the G p -action on X is versal for every prime p. Now (a) Conjecture 8.8 implies that X is G-versal. Thus ed(G) dim(X) 3.…”

Section: Projective Representationsmentioning

confidence: 88%

“…Remark 10.9. If we knew whether ed(PSL 2 (F 11 )) is 3 or 4, we would be able to complete the classification of finite simple groups of essential dimension 3 over C. For details, see [Bea11].…”

Section: Projective Representationsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

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Versality of algebraic group actions and rational points on twisted varieties

Duncan

Reichstein

2015

J. Algebraic Geom.

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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...

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Section: Projective Representationsmentioning

confidence: 88%

Section: Projective Representationsmentioning

confidence: 99%

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Versality of algebraic group actions and rational points on twisted varieties

Duncan

Reichstein

2015

J. Algebraic Geom.

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“…By Duncan and Reichstein [12, Lemma 10.1(c)], each T H[i] is a hypersurface of degree q si+1 in P(W K ). Hence, by Bezout's theorem [14,Proposition 8.4], deg K (Z) = deg( T H [1]) · · · deg( T H[n − d]) · deg(M 1 ) · · · deg(M d ) = q s1+1 · · · q s n−d +1 · 1 · · · 1 d times is a power of q, as desired.…”

Section: Proof Of Theorem 14: Preliminariesmentioning

confidence: 84%

“…On the other hand, by Beauville [1], ed(G) 4 for any of these groups, except for G PSL 2 (5) and (possibly) PSL 2 (11).…”

Section: A-groupsmentioning

confidence: 99%