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Invariant Rings and Quasiaffine Quotients

Joerg Winkelmann
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Abstract

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. Nevertheless one can prove that such a ring of invariants is always isomorphic to the ring of functions on a quasi-affine variety. Conversely, for a given quasi-affine variety V there exists always an action of the additive group on some affine variety W such that the ring of functions of V is isomorphic to the ring of invariant functions on W. Thus a k-algebra occurs as invariant ring for some group acting on a k-variety iff it occurs as function ring for some quasi-affine k-variety.

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Joerg Winkelmann. 2000-07-12. Invariant Rings and Quasiaffine Quotients. https://arxiv.org/abs/math/0007076

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