Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 313–327 | Cite as

Logarithmic Picard groups, chip firing, and the combinatorial rank

  • Tyler Foster
  • Dhruv Ranganathan
  • Mattia Talpo
  • Martin UlirschEmail author


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.

Mathematics Subject Classification

14H10 14T05 



The authors would like to thank Jonathan Wise for bringing Illusie’s logarithmic Picard group to their attention, sometimes through intermediaries, as well as Dan Abramovich, Yoav Len, and Jeremy Usatine for many discussions related to this topic. This collaboration started at the CMO-BIRS workshop on Algebraic, Tropical, and Non-Archimedean Analytic Geometry of Moduli Spaces; many thanks to the organizers Matt Baker, Melody Chan, Dave Jensen, and Sam Payne. Particular thanks are due to Farbod Shokrieh, for his advice to the last author concerning rank-determining sets (as hinted upon in Remark 5.2). The document has been improved by the helpful comments of an anonymous referee.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Tyler Foster
    • 1
  • Dhruv Ranganathan
    • 2
  • Mattia Talpo
    • 3
    • 4
  • Martin Ulirsch
    • 5
    Email author
  1. 1.Max Planck Institute for MathematicsBonnGermany
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Department of MathematicsSimon Fraser UniversityBurnabyCanada
  4. 4.Pacific Institute for the Mathematical SciencesVancouverCanada
  5. 5.Department of MathematicsUniversity of MichiganAnn ArborUSA

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