2000/07/12 by Joerg Winkelmann, Winkelmann, Joerg
Mathematics · #13A50 #14L30 #14R20 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:13A50 #msc:14L30 #msc:14R20
paper · pdf · doi:10.48550/arxiv.math/0007076
11 pages, LaTeX
arxiv created 2000/07/12 · arxiv updated 2009/11/30
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.