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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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