Residue polytopes

Abstract

A level graph is the data of a pair $(G,\ddot{I}€)$ consisting of a finite graph $G$ and an ordered partition $\ddot{I}€$ on the set of vertices of $G$. To each level graph on $n$ vertices we associate a polytope in ${\mathbb{R}}^n$ called its residue polytope. We show that residue polytopes are compatible with each other in the sense that if $\ddot{I}{€}^{\prime }$ is a coarsening of $\ddot{I}€$, then the polytope associated to $(G,\ddot{I}€)$ is a face of the one associated to $(G,\ddot{I}{€}^{\prime })$. Moreover, they form all the faces of the residue polytope of $G$, defined as the polytope associated to the level graph with the trivial ordered partition. The results are used in a companion work to describe limits of spaces of Abelian differentials on families of Riemann surfaces approaching a stable Riemann surface on the boundary of the moduli space.

18 pages, 3 figures