---
_id: '66338'
abstract:
- lang: eng
  text: 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.
author:
- first_name: Tyler
  full_name: Foster, Tyler
  last_name: Foster
- first_name: Dhruv
  full_name: Ranganathan, Dhruv
  last_name: Ranganathan
- first_name: Mattia
  full_name: Talpo, Mattia
  last_name: Talpo
- first_name: Martin
  full_name: Ulirsch, Martin
  id: '114697'
  last_name: Ulirsch
citation:
  ama: Foster T, Ranganathan D, Talpo M, Ulirsch M. Logarithmic Picard groups, chip
    firing, and the combinatorial rank. <i>arXiv:161110233</i>. Published online 2016.
  apa: Foster, T., Ranganathan, D., Talpo, M., &#38; Ulirsch, M. (2016). Logarithmic
    Picard groups, chip firing, and the combinatorial rank. <i>ArXiv:1611.10233</i>.
  bibtex: '@article{Foster_Ranganathan_Talpo_Ulirsch_2016, title={Logarithmic Picard
    groups, chip firing, and the combinatorial rank}, journal={arXiv:1611.10233},
    author={Foster, Tyler and Ranganathan, Dhruv and Talpo, Mattia and Ulirsch, Martin},
    year={2016} }'
  chicago: Foster, Tyler, Dhruv Ranganathan, Mattia Talpo, and Martin Ulirsch. “Logarithmic
    Picard Groups, Chip Firing, and the Combinatorial Rank.” <i>ArXiv:1611.10233</i>,
    2016.
  ieee: T. Foster, D. Ranganathan, M. Talpo, and M. Ulirsch, “Logarithmic Picard groups,
    chip firing, and the combinatorial rank,” <i>arXiv:1611.10233</i>, 2016.
  mla: Foster, Tyler, et al. “Logarithmic Picard Groups, Chip Firing, and the Combinatorial
    Rank.” <i>ArXiv:1611.10233</i>, 2016.
  short: T. Foster, D. Ranganathan, M. Talpo, M. Ulirsch, ArXiv:1611.10233 (2016).
date_created: 2026-07-08T07:08:39Z
date_updated: 2026-07-08T07:41:33Z
external_id:
  arxiv:
  - '1611.10233'
language:
- iso: eng
publication: arXiv:1611.10233
status: public
title: Logarithmic Picard groups, chip firing, and the combinatorial rank
type: journal_article
user_id: '82981'
year: '2016'
...