Divisors on graphs, Connected flags, and Syzygies

TL;DR

This paper explores the algebraic structure of ideals derived from divisors on graphs, providing explicit minimal Gr"obner bases and Betti number calculations that are characteristic-independent, extending previous work to general graphs.

Contribution

It offers a comprehensive description of minimal Gr"obner bases for syzygy modules of divisor ideals on graphs, generalizing prior results to all graphs.

Findings

Betti numbers match between binomial and initial ideals

Betti numbers are characteristic-independent

Explicit minimal Gr"obner bases are provided for all higher syzygy modules

Abstract

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gr\"obner theory. We give an explicit description of a minimal Gr\"obner bases for each higher syzygy module. In each case the given minimal Gr\"obner bases is also a minimal generating set. The Betti numbers of the binomial ideal and its natural initial ideal coincide and they correspond to the number of 'connected flags' in the graph. In particular the Betti numbers are independent of the characteristic of the base field. For complete graphs the problem was previously studied by Postnikov and Shapiro and by Manjunath and Sturmfels. The case of a general graph was stated as an open problem.