Secondary fans and secondary polyhedra of punctured Riemann surfaces

TL;DR

This paper extends the concept of secondary fans and polyhedra from finite point configurations to punctured Riemann surfaces, linking combinatorial decompositions of surfaces to convex polyhedral geometry.

Contribution

It introduces a new construction of secondary fans and polyhedra for punctured Riemann surfaces, analogous to the classical case of point configurations.

Findings

Secondary fan of Riemann surface is the normal fan of a convex polyhedron.

Cones correspond to ideal cell decompositions and horocyclic Delaunay decompositions.

Secondary polyhedron encodes combinatorial types of surface decompositions.

Abstract

A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration $A \subset \mathbb{R}^d$ a polyhedral fan, which stratifies the space of weight vectors by the combinatorial types of regular subdivisions of $A$. That fan arises as the normal fan of a convex polytope. In a completely analogous way we associate to each hyperbolic Riemann surface $R$ with punctures a polyhedral fan. Its cones correspond to the ideal cell decompositions of $R$ that occur as the horocyclic Delaunay decompositions which arise via the convex hull construction of Epstein and Penner. Similar to the classical case, this secondary fan of $R$ turns out to be the normal fan of a convex polyhedron, the secondary polyhedron of $R$.