Abstract. The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group.

Abstract. The well-known fiber dimension theorem in algebraic geometry says that for every morphism f : X → Y of integral schemes of finite type, the dimension of every fiber of f is at least dim X − dim Y . This has recently been generalized by P. Brosnan, Z. Reichstein and A. Vistoli to certain morphisms of algebraic stacks f : X → Y, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover we will prove a variant of this theorem, where essential dimension and canonical dimension are linked.These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by M. MacDonald, A. Meyer, Z. Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

We investigate the essential dimension of finite groups using the multihomogenization technique introduced in [KLS09], for which we provide new applications in a more general setting. We generalize the central extension theorem of Buhler and Reichstein [BR97, Theorem 5.3] and use multihomogenization as a substitute to the stackinvolved part of the theorem of Karpenko and Merkurjev [KM08] about the essential dimension of p-groups.

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 .

Abstract. Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G 0 is a torus.