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.
Mathematics Subject Classification (2000): 13A50, 14R20, 14L30
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