Voronoi tilings, toric arrangements and degenerations of line bundles III

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 article and the first two that preceded it are the first in a series aimed to explore this new approach.In Part I, we set up the combinatorial framework and showed how graphs weighted with integer lengths associated to the edges provide tilings of Euclidean spaces by certain polytopes associated to the graph itself and to its subgraphs.In Part II, we described the arrangements of toric varieties associated to the tilings of Part I in several ways: using normal fans, as unions of orbits, by equations and as degenerations of tori.In the present Part III, we show how these combinatorial and toric frameworks allow us to describe all stable limits of a family of line bundles along a degenerating family of curves. Our main result asserts that the collection of all these limits is parametrized by a connected 0-dimensional closed substack of the Artin stack of all torsion-free rank-one sheaves on the limit curve. Moreover, we thoroughly describe this closed substack and all the closed substacks that arise in this way as certain torus quotients of the arrangements of toric varieties of Part II determined by the Voronoi tilings of Euclidean spaces studied in Part I.

Lattice of Integer Flows and the Poset of Strongly Connected Orientations for Regular Matroids

“…In this section we extend this result to regular matroids, and show that it follows Theorem 3.13 via matroid duality. This theorem, in a more general context, later appeared in [AE1] and became an important tool in describing all stable limits of a family of line bundles along a degenerating family of curves [AE2,AE3]. 5.1.…”

“…Amini also proves a dual theorem for the lattice of integer cuts in [A]. This dual theorem later appeared in [AE1] in the context of toric geometry, as it is crucial in the study of the degeneration problem for linear series on curves [AE2,AE3]. Note that the framework in [AE1] is more general: the result required for the geometric applications is a mixed setting where the tiling is given by a collection of Voroni polytopes associated to some subgraphs of G, which are determined by the arithmetic of the integer edge lengths, and the divisor.…”

A Clifford inequality for semistable curves

“…Each approach comes with its own advantages and limitations, and in each case there is a large body of literature. See, for example, [4,7,17,[23][24][25] and [2,22].…”