Secondary fans and secondary polyhedra of punctured Riemann surfaces
Abstract
A famous construction of Gelfand, Kapranov and Zelevinsky associates to each finite point configuration A ⊂ Rd 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.
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