Brill-Noether theory of squarefree modules supported on a graph

TL;DR

This paper explores the algebraic and geometric properties of squarefree modules on graphs, establishing analogies with classical algebraic geometry results like Riemann-Roch, Jacobian, gonality, and Clifford's theorem.

Contribution

It introduces a Riemann-Roch theorem and studies Jacobian, gonality, and Clifford's theorem for squarefree modules on graphs, extending algebraic geometry concepts to combinatorial structures.

Findings

Proved a Riemann-Roch theorem for modules on graphs

Analyzed the Jacobian and gonality of a graph

Established Clifford's theorem in this context

Abstract

We investigate the analogy between squarefree Cohen-Macaulay modules supported on a graph and line bundles on a curve. We prove a Riemann-Roch theorem, we study the Jacobian and gonality of a graph, and we prove Clifford's theorem.