By Hiroaki Hikikata
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Extra resources for Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2
Choose such a component, say C, and fix it once for all. We write the canonical divisor K on S as Κ = -aC Ν ~Y^atCt\ i=2 a, ai G Ζ, a > α; > 1 (for every i > 2), a > 2, where C and CVs are distinct curves. Then it suffices to prove that S is a (CB)surface and that the curves conntained in R = Supp(—K) constitute a ( C B ) . We can make use of the following important result. 2 (Enoki). Let X be a surface of class VIIo with b2 > 0. Then X is biholomorphic to an exceptional compactification 5 U ) ta )of an affine bundle over an elliptic curve, if and only if there exists a non-zero divisor D 2 on X with (D )x — 0.
References [Κ] Y . Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces, J. für die reine und angew. Math. 363 (1985), 1-46. [V] E. Viehweg, Weak positivity and the additivity of the Kodaira dimension for certain fiber spaces, in Algebraic and Analytic Varieties, (S. ), Advanced Studies in Pure Math. 1, Kinokuniya, Tokyo, and North-Holland, Amsterdam, 1983, 329-353. Noboru N A K A Y A M A Department of Mathematics Faculty of Science University of Tokyo Hongo, Tokyo, 113 Japan Algebraic Geometry and Commutative Algebra in Honor of Masayoshi N A G A T A pp.
94(1972), 597608.  H. B. Laufer: On minimally elliptic singularitied. Amer. J. , 99(1977), 1257-1295.  I. Nakamura: On surfaces of class V I I 0 with global spherical shells. Proc. , 59A(1983), 29-32.  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.  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.
Algebraic Geometry and Commutative Algebra. In Honor of Masayoshi Nagata, Volume 2 by Hiroaki Hikikata