The slice method for G-torsors

Abstract

The notion of a (G, N)-slice of a G-variety was introduced by P.I. Katsylo in the early 80's for an algebraically closed base field of characteristic 0. Slices (also known under the name of relative sections) have ever since provided a fundamental tool in invariant theory, allowing reduction of rational or regular invariants of an algebraic group G to invariants of a "simpler" group. We refine this notion for a G-scheme over an arbitrary field, and use it to get reduction of structure group results for G-torsors. Namely we show that any (G, N)slice of a versal G-scheme gives surjective maps H 1 (L, N) → H 1 (L, G) in fppf-cohomology for infinite fields L containing F. We show that every stabilizer in general position H for a geometrically irreducible G-variety V gives rise to a (G, N G (H))-slice in our sense. The combination of these two results is applied in particular to obtain a striking new upper bound on the essential dimension of the simply connected split algebraic group of type E 7 .

$E_8$, the most exceptional group

“…The distance between the upper and lower bounds is remarkable. In contrast, for the other simply connected exceptional groups over C, one knows by [119] and [115] that ed(G 2 ) = 3, 5 ≤ ed(F 4 ) ≤ 7, 4 ≤ ed(E 6 ) ≤ 8, and 7 ≤ ed(E 7 ) ≤ 11, which are all much closer. Determining ed(E 8 ) will require new techniques.…”

“…As in [165], using the Z/3-grading on e 8 from Example 4.4, we find that SL 9 /μ 3 acts transitively on certain 4-dimensional subspaces of This gives a surjection in Galois cohomology H 1 (F, N ) → H 1 (F, SL 9 /μ 3 ) for some subgroup N of SL 9 /μ 3 , and one can hope that analyzing this surjection would give insight into whether algebras of degree 9 and period 3 are crossed products. See [115] for more discussion of this general setup and [66] for examples where similar surjections are exploited.…”

$E_8$, the most exceptional group

“…The distance between the upper and lower bounds is remarkable. In contrast, for the other simply connected exceptional groups over C, one knows by [Mac14] and [LM15b] that ed(G 2 ) = 3, 5 ≤ ed(F 4 ) ≤ 7, 4 ≤ ed(E 6 ) ≤ 8, and 7 ≤ ed(E 7 ) ≤ 11, which are all much closer. Determining ed(E 8 ) will require new techniques.…”

“…This gives a surjection in Galois cohomology H 1 (F, N ) → H 1 (F, SL 9 /µ 3 ) for some subgroup N of SL 9 /µ 3 and one can hope that analyzing this surjection would give insight into whether algebras of degree 9 and period 3 are crossed products. See [LM15b] for more discussion of this general setup and [Gar09a] for examples where similar surjections are exploited.…”

Essential dimension of algebraic groups, including bad characteristic

“…Exceptional groups. Concerning exceptional groups, a series of papers [Lem04], [Mac14], [Mac13], [LM15] have led to the following upper bounds for exceptional groups: ed(F 4 ) ≤ 7, ed(E sc 6 ) ≤ 8, and ed(E sc 7 ) ≤ 11 if char k = 2, 3. (Here F 4 , E sc 6 and E sc 7 denote simple and simply connected groups of types F 4 , E 6 , and E 7 ; the displayed upper bounds are meant to be compared with the dimensions of 52, 78, and 133 respectively.…”