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Hypersurface simple K3 singularities. (English) Zbl 0733.14017
The author classifies simple hypersurface K3 singularities (X,x) defined by a non-degenerated polynomial $$f(z_ 0,z_ 1,z_ 2,z_ 3)$$ and he studies the minimal resolution $$\pi: (\tilde X,E)\to (X,x)$$ and the singularities on the exceptional divisor E.
A three-dimensional singularity (X,x) is a simple K3 singularity if (X,x) is quasi-Gorenstein and if the exceptional divisor E of any minimal resolution is a normal K3 surface, where a minimal resolution $$\pi: (\tilde X,E)\to (X,x)$$ is a proper morphism with only terminal singularities on $$\tilde X,$$ with $$\tilde X\simeq X\setminus \{x\}$$ and with $$K_{\tilde X}$$ nef with respect to $$\pi$$.
- The simple K3 singularities could be regarded as three-dimensional generalizations of simple elliptic singularities.
If the simple K3 singularity (X,x) is defined by a non-degenerated polynomial f(z), then $$(1,1,1,1)\in \Gamma(f)$$. The weight $$\alpha =\alpha(f)=(\alpha_ 1,\alpha_ 2,\alpha_ 3,\alpha_ 4)$$ of the quasi-homogeneous polynomial $$f_{\Delta_ 0}$$ associated to the face $$\Delta_ 0$$ containing (1,1,1,1) verifies $$\sum^{4}_{i=1}\alpha_ i =1$$. - Then to classify the simple K3 singularities we need to study the set $$W_ 4$$ of weights: $$W_ 4=\{\alpha \in {\mathbb{Q}}^ 4_+| \quad \sum^{4}_{i=1}\alpha_ i =1,\alpha_ 1\geq...\geq \alpha_ 4$$ and $$(1,1,1,1)\in Int(C(\alpha))\},$$ where $$C(\alpha$$) is the closed cone in $${\mathbb{R}}^ 4$$ generated by the set $$T(\alpha)=\{\nu \in {\mathbb{Z}}^ 4_ 0| \alpha.\nu =1\}.$$
The author shows that the cardinality of $$W_ 4$$ is 95, and for each weight $$\alpha$$ he gives a quasi-homogeneous f of weight $$\alpha$$ which defines a simple K3 singularity and such that $$\Delta_ 0=\Gamma (f)$$ is the convex hull of $$T(\alpha$$). Then he constructs a minimal resolution $$\pi: \tilde X\to X$$ using torus embedding: if the weight $$\alpha(f)=(p_ 1/p,...,p_ 4/p)$$, where $$p_ 1,...,p_ 4$$ are relatively prime integers, the filtered blow-up with weight $$(p_ 1,...,p_ 4)$$, $$\Pi: (V,F)\to ({\mathbb{C}}^ 4,0)$$, induces a minimal resolution of (X,x). In the last part the author shows that under some conditions on the nondegenerate polynomial f defining the simple K3 singularity (X,x), the type and the number of the singularities on E are determined by the weight $$\alpha(f)$$, independently of f.
Reviewer: M.Vaquie (Paris)

##### MSC:
 14J17 Singularities of surfaces or higher-dimensional varieties 14J28 $$K3$$ surfaces and Enriques surfaces 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry
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##### References:
  A. R. FLETCHER, Plurigenera of 3-folds and weighted hypersurfaces, thesis submitted for the degree of Doctor of Philosophy at the University of Warwick, 1988.  S. ISHII, Onisolated Gorenstein singularities, Math. Ann. 270(1985), 541-554 · Zbl 0541.14002 · doi:10.1007/BF01455303 · eudml:182978  S. ISHII, On the classification of two dimensional singularities by the invariant K, preprint, 1988  S. ISHII AND K. WATANABE, On simple K3 singularities(in Japanese), Notes appearing in the Proceeding of the Conference on Algebraic Geometry at Tokyo Metropolitan Univ, 1988, 20-31.  M. -N. ISHIDA AND N. IwASHiTA, Canonical cyclic quotient singularities of dimension three, in Comple Analytic Singularities (T. Suwa and P. Wagreich, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1986, 135-151. · Zbl 0627.14002  G. KEMPF, F. KNUDSEN, D. MUMFORD AND B. SAINT-DONAT, Toroidal Embeddings I, Lecture Note in Math. 339, Springer-Verlag, Berlin Heidelberg, New York, 1973. · Zbl 0271.14017  D. R. MORRISON AND G. STEVENS, Terminal quotient singularitiesin dimensions three and four, Proc Amer. Math. Soc. 90 (1984), 15-20. · Zbl 0536.14003 · doi:10.2307/2044659  T. ODA, Lectures on Torus Embeddings and Applications (Based on joint work with Katsuya Miyake), Tata Inst. of Fund. Reserch 58, Splinger-Verlag, Berlin-Heiderberg-New York, 1978. · Zbl 0417.14043  M. OKA, On the resolution of hypersurface singularities, in Complex Analytic Singularities (T. Suw and P. Wagreich, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1986, 405-436. · Zbl 0622.14012  M. REID, Canonical 3-folds, Journees de Geometric algebrique d’Angers, (A. Beauville, ed.), Sijthof and Noordhoff, Alphen aan den Rijn, 1980, 273-310. · Zbl 0451.14014  K. SAITO, Einfach-elliptische Singularitaten, Invent. Math. 23 (1974), 289-325 · Zbl 0296.14019 · doi:10.1007/BF01389749 · eudml:142265  M. TOMARI, The canonical filtration of higher dimensional purely elliptic singularity of a special type, preprint, 1989. · Zbl 0778.14014 · doi:10.1007/BF01245087 · eudml:143893  M. TOMARI AND K. -i. WATANABE, Filtered rings, filtered blowing-ups and normal two-dimensiona singularities with ”star-shaped” resolution, Submitted to Publ. Res. Inst. Math. Sci., Kyoto Univ. · Zbl 0725.13003 · doi:10.2977/prims/1195172704  A. N. VARCHENKO, Zeta-Function of monodromy and Newton’s diagram, Invent. Math. 37 (1976), 253-262. · Zbl 0333.14007 · doi:10.1007/BF01390323 · eudml:142438  K. WATANABE, On plurigenera of normal isolated singularities, I, Math. Ann. 250 (1980), 65-94 · Zbl 0414.32005 · doi:10.1007/BF02599787 · eudml:163411  K. WATANABE, On plurigenera of normal isolated singularities, II, in Complex Analytic Singularitie (T. Suwa and P. Wagreich, eds.), Advanced Studies in Pure Math. 8, Kinokuniya, Tokyo and North-Holland, Amsterdam, New York, Oxford, 1986, 671-685. · Zbl 0659.32015  K. WATANABE AND T. YONEMURA, On ring theoretic genus pr and plurigenera m of normal isolate singularities, preprint, 1988.
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