Algebraic Geometry and Commutative Algebra. In Honor of by Hiroaki Hijikata

By Hiroaki Hijikata

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Extra info for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 1

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This allows us to apply Nikulin's theorem [N] to the sixteen disjoint rational curves M¿, i = 1, . . ) The reader will reahze that the curve S is invariant imder an action of the icosahedral group on P3. Although elementary, this fact is not so easy to check, cf. 1]. Indeed the parameter λ on 5 is the "Ikosaeder Transzen­ dente" of F. Klein parametrizing elUptic curves with a level-5 structure. The polynomial identities needed are related to the modular equation (the polyno­ mial P3 in Section 1 below) describing 2-torsion quotients of such curves.

Ai)Ap : b holds for 0 < ¿ < d (cf. 18]). For every prime ideal p 2 H\D*)p we have =0 for ¿ > 0 as Ap is Cohen-Macaulay. Hence there is a positive integer t such that PHI^{A) = 0 for i < d. Let p and q be prime ideals such that p 2 ^ and q 2 / + p. Since Ap is Cohen-Macaulay and height q > d, depth Ap + dim Aq/pAq = dim Ap -f dim Aq/pAq = dim Aq > d. Hence there is a positive integer u such that Ι^Η}{Α) = 0 for i < d by virtue of [6]. Taking powers of x i , . . ,Xd again if necessary, we may assume that all of x i , .

31 A Conjecture of Sharp =3 m^E + Ä+ for ¿ = 1 , . . -n Let R = R{A,J), ... = mi? -f generated A-algebra). = Note that R has a duaUzing complex (as Ä is a finitely T h e o r e m 3 . 1 . , 0^, (cf. 17] and [15,§1]). We put p = φη A. First suppose p is not a maximal ideal. If p 2 then Rp = Ap[T] is CohenMacaulay. 19]. Hence Rφ is Cohen-Macaulay. Now suppose p is a maximal ideal. Considermg Ap and Rp instead of A and R respectively, we may assume that A is a local ring. , yx^^)r-' and φ 2 Φ ^ x^T or φ ^ yx^T for some i.

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