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By Hiroaki Hikikata

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

**Example text**

As ΒΊ is a Krull domain, the number of essential valuations ν in ΒΊ with υ(χ) φ 0, corresponding to A s s ( 5 7/ x 5 7) , is finite for any 7 G Γ ( 7 ι ) . Then, Δ(χ) being a finite set, one find 72 G Γ ( 7 ι ) such that A s s ( 5 7 / x Ë 7 ) —> Ass(By/xBy) is surjective for any 7 ' D 7 ( D 73). Hence, there is 72 G Γ ^ ) such that Ass(Bj/xBy) —> Ass(j§y/χΒ Ί>) is bijective for any 7 ' D 7 ( D 72). Let { α ; ι , . . , ω Γ } be the set of essential valuations ω in # 7 2 with ω(χ) φ 0 and Wi be the DVR associated to U{.

It follows that Ä is a (CB) and 5 is a (CB)-surface. d. The following proposition describes a singularity obtained by blowing-down of curves on a (CB)-surface. 6. Let S be a (CB)-surface and R its ( C B ) . We assume that Ks — —D, where D is an effective divisor with SuppjD = Suppi?. Let φ: S —• 5o be the contraction of R to a point x. Then the point χ is a Gorenstein singularity with geometric genus two. 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.

8] I. Nakamura: On surfaces of class V I I 0 with global spherical shells. Proc. , 59A(1983), 29-32. [9] I. Nakamura: On surfaces of class VIIo with curves. Inv. , 78(1984), 393-443. [9 bis] I. Nakamura: On surfaces of class VIIo with curves, I I . Preprint. [10] K . Nishiguchi: Degeneration of surfaces with trivial canonical bundles. Proc. , 59A(1983), 304-307. [II] K . Nishiguchi: Canonical bundles of compact complex surfaces containing global spherical shells. Proc. , 62A(1986), 234-237. [12] K .