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

By Hiroaki Hikikata

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Extra resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2

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88(1966), 129-136. [2] G. Dloussky: Une propriété topologique des surfaces de Kato. Preprint. [3] I. Enoki: Surfaces of class V I I 0 with curves. Tohoku Math. , 33(1981), 453-492. [4] Ma. Kato: Compact complex manifolds containing global spherical shells, I. Proc. Int. Symp. on Algebraic Geometry (Kyoto), Kinokuniya, Tokyo, pp. 45-84(1977). [5] K . Kodaira: On the structure of compact complex analytic surfaces, I, I I , III, I V . Amer. J. , 86(1964), 751-798; 88(1966), 682-721; 90(1968), 55-83; 90(1968), 170-192.

1. Remark. 1 is shown in a different way. §3. 1. The result will be used later for the classification of certain degenerations of K3 surfaces (cf. Nishiguchi[10,12]). 1. Let S be a surface of class V I I Q . Suppose that the canonical bundle Κ of S is written as Κ = —D, where D is a non-zero reduced effective divisor. Then S is one of the following: (a) a Hopf surface, (b) a parabolic Inoue surface, (c) a hyperbolic Inoue surface. Proof. 1, we may assume that a(S) = 0, 62 {S) > 0. 4. 2 (iii), it is enough to consider the case where each irreducible component of D is either a non-singular rational curve or a rational curve with a node.

6). Take a minimal prime ideal Ρ of χ A. As C is finite over A , we find Q G A s s ( C / x C ) such that Q Π A = P. This means P G Δ(χ) (cf. 1)). Hence 1 = ht Ρ > ht p 7 + 1 for any 7 G Γ ( 7 2 ) . Therefore, ht p* = 0. 6) is proved and this completes the proof of Rotthaus' Hilfssatz. Next proposition shows that, in some practical cases, a bounded prime ideal sequence automatically becomes good. 18. Let A be a noetherian ring and m G M a x ( A ) . Fixing an element x G A and a minimal prime ideal pj^ of 6p(A^n) ( = Spec(A^) Kr(A*m)), let p; = p*m Π A; for 7 e r ( m ) ( = r ( { m } ) ) .

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