Combinatorial Properties of the K3 Surface: Simplicial Blowups and Slicings

The 4-dimensional abstract Kummer variety K4 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 the minimal 16-vertex triangulation (K)16 we resolve its 16 isolated singularities – step by step – by simplicial blowups. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the 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 2000: Primary 57Q15; Secondary 14J28, 14E15, 57Q25, 52B70