Invariant Rings and Quasiaffine Quotients

Jörg Winkelmann

We study Hilbert's fourteenth problem from a geometric point of view. Nagata's celebrated counterexample demonstrates that for an arbitrary group action on a variety the ring of invariant functions need not be isomorphic to the ring of functions of an affine variety. In this paper we will show that nevertheless it is always isomorphic to the ring of functions on a quasi-affine variety.

Actually we show that, given a field k and a integrally closed k-algebra R the following conditions are equivalent:

• There exists an irreducible, reduced k-variety V and a subgroup G of Aut_k(V) such that R is isomorphic to the ring of G invariant k-regular functions on V.
• There exists a quasi-affine irreducible, reduced k-variety V such that R is isomorphic to the ring of k-regular functions on V.
• There exists an affine irreducible, reduced k-variety V and a regular action of the additive group G_a on V defined over k such that R is isomorphic to the ring of G_a invariant k-regular functions on V.

Math. Z. 244, 163-174 (2003)
DOI: 10.1007/s00209-002-0484-9