Birationally isotrivial fiber spaces

We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.Comment: 10 pages; v2: thanks to helpful suggestions of the referee, the main result is now independent of the Eisenbud-Goto conjecture; also we removed the erroneous Prop. 3.4 of v1 which was not needed for anything else in the pape

Low-dimensional Singularities with Free Divisors as Discriminants

We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C^5, namely the simple elliptic singularity of type \tilde A_4, we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j-invariant is the cone over the dual variety of an elliptic curve.Comment: 29 pages, misprints and references correcte

Rationality of quotients by linear actions of affine groups

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.Comment: 18 pages; dedicated to Fabrizio Catanese on the occasion of his 60th birthda