A tessellation for Fermat surfaces in CP~3

摘要

For each positive integer n, we present a tessellation of CP~2 that can be lifted, through the branched covering, to a symmetric tessellation of the Fermat surface (a 4-manifold) of degree n in CP~3. The process is systematic and symbolically algebraic. Each four-cell in the tessellation is bounded by four pentahedrons, and each pentahedron has four triangular faces and one quadrilateral face. Graphically, one can produce the entire surface from one single four-cell using translations generated by permutations and phase multiplications of the homogeneous coordinates of CP~3. Note that the tessellation of the Fermat surface of degree 4, a K3 surface, has exactly 24 vertices.