Representations of SL2 and the distribution of points in ℙn
In 1958 Nagata  gave an ingenious argument that demonstrated the existence of counterexamples to Hilbert’s Fourteenth Problem. Recall that the original problem is the following: Let K = F(x 1, x 2,..., x n ) be the function field of affine n-space V = F n over an algebraically closed field F, and suppose L ⊂ K is any subfield. Then the question is: Is A = L ∩ F[x 1 , x 2 ,..., xn] a finitely generated F-algebra. In most cases of interest, L is the field of invariants of an algebraic group G acting linearly on V, and A becomes the ring of invariant regular functions on V. Certainly, if G is reductive, the answer is yes, a result due to Hilbert himself for char(F) = 0, but much more subtle in positive characteristic (see ). More generally for G reductive O(X) G is a finitely generated F-algebra for any affine variety X over F on which G acts.
KeywordsGeneric Point Algebraic Group Transformation Rule Formal Power Series Hilbert Series
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- R. Steinberg, Nagata’s example, Algebraic groups and Lie groups, Austral. Math. Soc. Lect. Ser., 9, Cambridge Univ. Press, Cambridge, 1997, pp. 375–384.Google Scholar