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By Stouffer E. B.

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Let P be the variety 1 ;1 obtained from L; D by adding the point at infinity in each fibre of L D . More precisely, let X be the trivial line bundle. 2. THE WEITZENBOCK THEOREM Then we obtain that R (D ) is equal to the ring R(S ) where infinity in P . In this way we are led to the following. 49 S is the divisor at Problem 2. (O. Zariski) Let X be a nonsingular algebraic variety and let an effective divisor on X . When is the algebra R (D ) finitely generated? 3). It turns out that the algebras R (D ) are often not finitely generated.

We obtain that A = (S=I )G . 5, AG is integral over S G =I S G . Since we have shown already that S G is finitely generated, we are almost done (certainly done in the case when G is linearly reductive). By a previous case we may assume that AG has no zerodivisors. 3) gives that the integral closure R of S G =I S G in the field of fractions Q(AG ) of AG is a finitely generated k -algebra provided that Q(AG ) is a finite extension of the field of fractions of S G =I S G . 2). Thus it is enough to show that the field Q(A) is a finite extension of the field of fractions of S G =I S G .

3. , I so that A = k T0 : : : Tn ] with grading defined by Ti Aqi , the set 2 q0 : : : qn) = Projm(A) = P( A n+1 = f0g), n f0g =k is called the weighted projective space with weights q 0 : : : qn . When all the qi are equal to 1, we obtain the usual definition of the n-dimensional projective space Pn (k ). Let n be the closed subgroup of G m = Spm(k T T ;1 ]) defined by the ideal n (T 1). As an abstract group it is isomorphic to the group of nth roots of 1 in k. Let A be a graded k-algebra and G m Aut(A) be the corresponding action.

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