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On the singular K-3 surfaces with hypersurface singularities. (English) Zbl 0609.14020

Let A be a projective algebraic surface over the complex number field \({\mathbb{C}}\). Assume that every singularity on A is a hypersurface isolated singularity, the dualizing sheaf of A is a trivial line bundle, \(H^ 1(A,{\mathcal O}_ A)=0\) and \(\dim (H^ 2(A,{\mathbb{R}}))=1\). Let \(M\to A\) denote the minimal resolution of singularities on A. The author shows that M is birationally equivalent to a ruled surface over a curve with a genus \(q,\) \(0\leq q\leq 3\), and that if \(q>0\), then the configuration of exceptional curves on M is one of the 16 possibilities which are given explicitly in this article.
Such surfaces are related to the compactification of \({\mathbb{C}}^ 3\). Let X be a compact Kähler smooth variety of dimension \(3.\) If X contains a normal analytic subset A such hat \(X-A\cong {\mathbb{C}}^ 3\), then the canonical line bundle of X is isomorphic to \({\mathcal O}_ X(-rA)\) with \(1\leq r\leq 4\). Moreover if \(r=1\), then A satisfies the above conditions [L. Brenton and J. Morrow, Trans. Am. Math. Soc. 246, 139-153 (1978; Zbl 0416.32015)]. Therefore we can regard this article as one step to determine the compactification X of \({\mathbb{C}}^ 3\). - The author’s method used to deduce the bound \(q\leq 3\) is very interesting. However, to tell the truth, this article is hard to read. It contains unnecessary parts and does not contain necessary explanations. For example, lemma U 1 on page 69, lemma 1 on page 70, and the expression on page 71 with beginning ”Since \(H^ 0(U,{\mathcal O}_ Z)...''\) and ending ”\(\chi\) (Z)\(\leq 0''\) are not necessary. In corollary 2, why the case \(q=1\) and \(b_ 2(M)=9\) is excluded should be remarked. In \(proposition\quad 6\) when \(q=2\) it should be explained why the cases \(Z\cdot Z=-4\) and \((e,s)=(-2,3), (-1,2)\); \(Z\cdot Z=-3\) and \((e,s)=(-2,2), (-1,1)\) are excluded. In table I item (8) the number 2 should be replaced by 1, and in item (12) in the same table we should add [1] below the circle with the number 3. Finally we cannot find the item Yau [14] in the reference list.
As it is explained in this article, the compactification X of \({\mathbb{C}}^ 3\) is a Fano 3-fold, which was studied by V. A. Iskovskikh [Math. USSR, Izv. 12, 469-506 (1978); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 42, 506-549 (1978; Zbl 0407.14016)]. If (X,A) is a compactification of \({\mathbb{C}}^ 3\), then \(b_ 0(X)=b_ 2(X)=b_ 4(X)=b_ 6(X)=1\), \(b_ 1(X)=b_ 5=0\), \(b_ 3(X)=2q\), \(0\leq q\leq 3\).
Reviewer: T.Urabe

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
32J15 Compact complex surfaces
14J15 Moduli, classification: analytic theory; relations with modular forms
14B05 Singularities in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J30 \(3\)-folds
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