Hanspeter KRAFT

Let [latex]G[latex] be a finite group and [latex]V[latex] a [latex]G[latex]-variety, i.e. an irreducible algebraic variety with a regular action of [latex]G[latex]. A compression of [latex]V[latex] is a [latex]G[latex]-equivariant dominant morphism [latex]f : Vto X[latex] such that [latex]G[latex] acts faithfully on [latex]X[latex]. The basic questions are : (a) How much can one compress a given action ? (b) What are the incompressible [latex]G[latex]-varieties ?

We first discuss this concept from two rather different points of view : (i) Generic struc- ture of Galois-coverings and (ii) Equations for field extension. We then define the covariant dimension of [latex]G[latex] which measures how much a representation of [latex]G[latex] can be compressed. This has to be compared with the essential dimension of [latex]G[latex] which was introduced by Buehler and Reichstein in order to study the number of parameters of equations. Finally, we will give a short overview on known results, work out a few interesting examples and discuss some open questions. (This is mostly joint work with G.W. Schwarz.)