Voronoi tilings, toric arrangements and degenerations of line bundles I

TL;DR

This paper introduces a combinatorial and geometric framework using Voronoi tilings and toric arrangements to understand limits of line bundles on degenerating curves, proposing a new approach to limit linear series.

Contribution

It establishes a novel connection between Voronoi tilings, toric arrangements, and degenerations of line bundles, setting the foundation for a new theory of limit linear series.

Findings

Describes how graphs with integer edge lengths induce Euclidean tilings.

Provides a detailed combinatorial structure of the associated polytopes.

Lays groundwork for analyzing stable limits of line bundles in degenerating families.

Abstract

We describe limits of line bundles on nodal curves in terms of toric arrangements associated to Voronoi tilings of Euclidean spaces. These tilings encode information on the relationship between the possibly infinitely many limits, and ultimately give rise to a new definition of limit linear series. This paper and its second and third companion parts are the first in a series aimed to explore this new approach. In the present article, we set up the combinatorial framework and show how graphs with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to certain of its subgraphs. We further provide a description of the combinatorial structure of these polytopes and the way they are glued together in the tiling. In the second part of the series, we describe the arrangements of toric varieties associated to these tilings. These results will be of use in the third part to achieve our goal of describing all stable limits of a family of line bundles along a degenerating family of curves.