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Combinatorial properties of the \(K3\) surface: simplicial blowups and slicings. (English) Zbl 1279.57018

Authors’ abstract: The 4-dimensional abstract Kummer variety \(K^4\) with 16 nodes leads to the \(K3\) surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of \(K^4\), we resolve its 16 isolated singularities – step by step – by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL \(K3\) surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover, we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the \(K3\) surface of various topological types.

MSC:

57Q15 Triangulating manifolds
14J28 \(K3\) surfaces and Enriques surfaces
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
57Q25 Comparison of PL-structures: classification, Hauptvermutung
52B70 Polyhedral manifolds

Software:

simpcomp
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