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.
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References
Amini, O., Baker, M.: Linear series on metrized complexes of algebraic curves. Math. Ann. 362(1–2), 55–106 (2015)
Amini, O., Caporaso, L.: Riemann-Roch theory for weighted graphs and tropical curves. Adv. Math. 240, 1–23 (2013)
Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M., Sun, S.: Logarithmic geometry and moduli, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM), vol. 24, pp. 1–61. Int. Press, Somerville (2013)
Artin, M., Grothendieck, A., Verdier, J.-L.: Théorie des topos et cohomologie etale des schémas. Lecture Notes in Mathematics, vol. 305. Springer, Berlin (1971)
Baker, M.: Specialization of linear systems from curves to graphs. Algebra Numb. Theory 2(6), 613–653 (2008) (With an appendix by Brian Conrad)
Baker, M., Jensen, D.: Degeneration of linear series from the tropical point of view and applications, Non-archimedean and tropical geometry. In: Baker, M., Payne, S. (eds.) Simons Symposia, vol. 1, pp. 365–433. Springer, New York (2016)
Baker, M., Norine, S.: Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)
Caporaso, Lucia: A compactification of the universal Picard variety over the moduli space of stable curves. J. Am. Math. Soc. 7(3), 589–660 (1994)
Caporaso, L.: Néron models and compactified Picard schemes over the moduli stack of stable curves. Am. J. Math. 1–47 (2008)
Cavalieri, R., Chan, M., Ulirsch, M., Wise, J.: A Moduli Stack of Tropical Curves. arXiv:1704.03806 [math] (2017)
Eisenbud, D., Harris, Joe: Limit linear series: basic theory. Invent. Math. 85(2), 337–371 (1986)
Foster, T.: Introduction to Adic Tropicalization. arXiv:1506.00726 (2015)
Foster, T., Payne, S.: Limits of Tropicalizations II: Structure Sheaves, Adic Spaces and Cofinality of Gubler Models. In preparation (2017)
Foster, T., Ranganathan, D.: Hahn analytification and connectivity of higher rank tropical varieties. Manuscripta Mathematica 1–22 (2015)
Hladký, J., Král’, D., Norine, S.: Rank of divisors on tropical curves. J. Combin. Theory Ser. A 120(7), 1521–1538 (2013)
Illusie, L.: Logarithmic spaces (according to K. Kato), Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991), Perspect. Math., vol. 15, pp. 183–203. Academic, San Diego (1994)
Kajiwara, T.: Logarithmic compactifications of the generalized Jacobian variety. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 40(2), 473–502 (1993)
Kato, K.: Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), pp. 191–224. Johns Hopkins Univ. Press, Baltimore (1989)
Kato, F.: Log smooth deformation theory. Tohoku Math. J. (2) 48(3), 317–354 (1996)
Kato, F.: Log smooth deformation and moduli of log smooth curves. Int. J. Math. 11(2), 215–232 (2000)
Len, Y.: Hyperelliptic graphs and metrized complexes. Forum Math. Sigma 5, e20, 15 (2017)
Luo, Y.: Rank-determining sets of metric graphs. J. Combin. Theory Ser. A 118(6), 1775–1793 (2011)
Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and theta functions, curves and abelian varieties, Contemp. Math., vol. 465, pp. 203–230. Am. Math. Soc., Providence, RI (2008)
Olsson, M.C.: Semistable degenerations and period spaces for polarized \(K3\) surfaces. Duke Math. J. 125(1), 121–203 (2004)
Olsson, M.C.: (Log) twisted curves. Compos. Math. 143(2), 476–494 (2007)
Parker, B.: Log geometry and exploded manifolds, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, vol. 82, pp. 43–81. Springer, New York (2012)
Raynaud, M.: Spécialisation du foncteur de Picard. Publications Mathématiques de l’IHÉS 38, 27–76 (1970)
Ulirsch, M.: Functorial tropicalization of logarithmic schemes: the case of constant coefficients. Proc. Lond. Math. Soc. (3) 114(6), 1081–1113 (2017)
Acknowledgements
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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T.F.’s research was supported by Institut des Hautes Études Scientifiques, by Le Laboratoire d’Excellence CARMIN, and by a Postdoctoral Fellowship at Max Planck Institute for Mathematics. M.T. was supported by a Postdoctoral Fellowship at the University of British Columbia. M.U.’s research was supported in part by funds by the SFB/TR 45 ’Periods, Moduli Spaces and Arithmetic of Algebraic Varieties’ of the DFG (German Research Foundation), the Hausdorff Center for Mathematics at the University of Bonn, and the Fields Institute for Research in Mathematical Sciences, Toronto.
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Foster, T., Ranganathan, D., Talpo, M. et al. Logarithmic Picard groups, chip firing, and the combinatorial rank. Math. Z. 291, 313–327 (2019). https://doi.org/10.1007/s00209-018-2085-2
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DOI: https://doi.org/10.1007/s00209-018-2085-2