A tessellation for algebraic surfaces in CP3

TL;DR

This paper introduces an explicit, symmetric tessellation algorithm for algebraic surfaces F(n) in complex projective 3-space, minimizing vertices and leveraging group actions, with applications to surfaces like K3.

Contribution

It provides the first explicit tessellation method for F(n) surfaces in CP3, using minimal vertices and symmetry properties, extending previous curve tessellations in CP2.

Findings

Tessellation uses n-th roots of unity in CP1 for minimal vertices

The tessellation is invariant under surface symmetries

Applicable to complex surfaces including K3 in CP3

Abstract

We present an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation z0^n + z1^n + z2^n + z3^n = 0 in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Our tessellation contains a minimal number of vertices, namely the n-th roots of unity in the six standard projective lines CP1 in CP3, which form an obvious framework for constructing a natural tessellation of F(n). Our tessellation is invariant under the action of the obvious isomorphism group of F(n) induced by permutations and phase multiplications of the coordinates, and the action is transitive on the set of 4-cells. The tessellation is built upon a similar triangulation for the corresponding algebraic curves in CP2.