Logarithmic Picard groups, chip firing, and the combinatorial rank

TL;DR

This paper explores the logarithmic Picard group of curves, connecting it to chip firing and graph divisor theory, and introduces a combinatorial rank with a Riemann-Roch analogue.

Contribution

It introduces a new perspective on the logarithmic Picard group, linking it to chip firing relations and defining a combinatorial rank with a Riemann-Roch theorem.

Findings

Logarithmic Picard group as a quotient of the algebraic Picard group

Definition of a combinatorial rank for line bundles

An analogue of the Riemann-Roch formula for the combinatorial rank

Abstract

Illusie has suggested that one should think of the classifying group of $M_X^{gp}$-torsors on a logarithmically smooth curve $X$ over a standard logarithmic point as a logarithmic analogue of the Picard group of $X$. This logarithmic Picard group arises naturally as a quotient of the algebraic Picard group by lifts of the chip firing relations of the associated dual graph. We connect this perspective to Baker and Norine's theory of ranks of divisors on a finite graph, and to Amini and Baker's metrized complexes of curves. Moreover, we propose a definition of a combinatorial rank for line bundles on $X$ and prove that an analogue of the Riemann-Roch formula holds for our combinatorial rank. Our proof proceeds by carefully describing the relationship between the logarithmic Picard group on a logarithmic curve and the Picard group of the associated metrized complex. This approach suggests a natural categorical framework for metrized complexes, namely the category of logarithmic curves.