378 research outputs found
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
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