Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Residue polytopes

Omid Amini, Eduardo Esteves, Eduardo Garcez

Abstract

A level graph is the data of a pair $(G,π)$ consisting of a finite graph $G$ and an ordered partition $π$ 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 $π'$ is a coarsening of $π$, then the polytope associated to $(G,π)$ is a face of the one associated to $(G,π')$. 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.

Residue polytopes

Abstract

A level graph is the data of a pair consisting of a finite graph and an ordered partition on the set of vertices of . To each level graph on vertices we associate a polytope in called its residue polytope. We show that residue polytopes are compatible with each other in the sense that if is a coarsening of , then the polytope associated to is a face of the one associated to . Moreover, they form all the faces of the residue polytope of , 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.

Paper Structure

This paper contains 17 sections, 15 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Abelian differentials on Riemann surfaces and their limits
  3. Set-theoretically independent collections
  4. Imposing the first three conditions
  5. Vanishing residue along downward arrows
  6. Local residue conditions
  7. Rosenlicht residue conditions
  8. Global residue conditions and proof of Theorem \ref{['thm:dimG']}
  9. Proof of Theorem \ref{['thm:dimG']}
  10. An example
  11. Splitting, realization and degeneration
  12. Submodular functions of subspaces
  13. Filtration associated to an ordered partition
  14. Splitting
  15. Realization
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\pi$ be an ordered partition of $V$. We have

Figures (3)

  • Figure 1: A level graph $(G,\pi)$ with two levels, $\pi=( \pi \sp{} !_1, \pi \sp{} !_2)$ with $\pi \sp{} !_1=\{ u \sp{} !_4, u \sp{} !_5\}, \pi \sp{} !_2 = \{ u \sp{} !_1, u \sp{} !_2, u \sp{} !_3\}$. The arrows $u \sp{} !_1 u \sp{} !_4, u \sp{} !_1 u \sp{} !_5, u \sp{} !_2 u \sp{} !_4, u \sp{} !_2 u \sp{} !_5, u \sp{} !_3 u \sp{} !_5$ are upward. The edge $\{ u \sp{} !_2, u \sp{} !_3\}$ is horizontal.
  • Figure 2: The residue polytope of the complete graph $K_4$ on four vertices on the right. The figure on the left shows the position of $\mathbf P \sp{} !_{ \mathrm{res} \sp{} !}(K_4)$ within the simplex $OXYZ$ of width three in $\mathbb R \sp{} !^4$, with vertices $O=(3,0,0,0)$, $X=(0,3,0,0)$, $Y=(0,0,3,0)$ and $Z=(0,0,0,3)$, using the projection to $\mathbb R \sp{} !^3$ given by the last three coordinates.
  • Figure 3: A level graph with three levels. Summits are drawn in red. There are three irreducible and two reducible summits. The subgraph $G[\{u_9,u_a\}]$ is a special connected component in $\mathcal{C}\mathcal{C} \sp{} !_{h<3}$, but it is nonspecial in $\mathcal{C}\mathcal{C} \sp{} !_{h<2}$.

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 19 more