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Sektion 11
Montag, 18.09.2000, 15.00–15.20 Uhr, POT 351

Invariant Rings and Quasiaffine Quotients

Jörg Winkelmann, Universität Basel

Let G be an algebraic group acting on an affine variety V . Hilbert’s 14th problem asked whether in this situation the ring of invariants must be finitely generated. Nagata showed that this is not always the case. Thus the ring of invariants may not be isomorphic to the ring of functions on an affine variety. Nevertheless it is necessarily isomorphic to the ring of functions of an quasi-affine variety. More precisely we show: Let k be a field and R an integrally closed k-algebra.

Then the following properties are equivalent:

  1. There exists an irreducible, reduced k-variety V and a subgroup G < Autk(V ) such that R  -~ k[V ]G.
  2. There exists a quasi-affine irreducible, reduced k-variety V such that R  -~ k[V ].
  3. There exists an affine irreducible, reduced k-variety V and a regular action of Ga = (k, +) on V defined over k such that R  -~ k[V ]Ga.

See also:
http://www.cplx.ruhr-uni-bochum.de/~jw/papers/hilbert14.html