Divisors on graphs, Connected flags, and Syzygies

Abstract

We study the binomial and monomial ideals arising from linear equivalence of divisors on graphs from the point of view of Gröbner theory. We give an explicit description of a minimal Gröbner bases for each higher syzygy module. In each case the given minimal Gröbner 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.

to appear in International Mathematics Research Notices (IMRN)

References in corpus (5)

• Chip-firing games, potential theory on graphs, and spanning trees
• Minimal Free Resolutions of the $G$-parking Function Ideal and the Toppling Ideal
• Wilmes' Conjecture and Boundary Divisors
• Primer for the algebraic geometry of sandpiles
• Laplacian ideals, arrangements, and resolutions

Cited by in corpus (10)

• Prime splittings of Determinantal Ideals
• Divisors on graphs, binomial and monomial ideals, and cellular resolutions
• Fourientations and the Tutte Polynomial
• Minimal Free Resolutions of the $G$-parking Function Ideal and the Toppling Ideal
• Wilmes' Conjecture and Boundary Divisors
• Another proof of Wilmes' conjecture
• Minimal cellular resolutions of powers of matching field ideals
• Laplacian ideals, arrangements, and resolutions
• Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions
• A special case of Postnikov-Shapiro conjecture