Combinatorial Properties of the K3 Surface: Simplicial Blowups and Slicings

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.

2000 AMS Subject Classification:

• Primary 57Q15
• Secondary 14J28
• 14E15
• 57Q25
• 52B70

Keywords:

• combinatorial manifold
• combinatorial pseudomanifold
• intersection form
• K3 surface
• Kummer variety
• resolution of singularities
• simplicial Hopf map

ACKNOWLEDGMENTS

This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG) under the grant Ku 1203/5-2.