Mathematische Annalen

, Volume 362, Issue 1–2, pp 55–106 | Cite as

Linear series on metrized complexes of algebraic curves

  • Omid Amini
  • Matthew BakerEmail author


A metrized complex of algebraic curves over an algebraically closed field \(\kappa \) is, roughly speaking, a finite metric graph \(\Gamma \) together with a collection of marked complete nonsingular algebraic curves \(C_v\) over \(\kappa \), one for each vertex \(v\) of \(\Gamma \); the marked points on \(C_v\) are in bijection with the edges of \(\Gamma \) incident to \(v\). We define linear equivalence of divisors and establish a Riemann–Roch theorem for metrized complexes of curves which combines the classical Riemann–Roch theorem over \(\kappa \) with its graph-theoretic and tropical analogues from Amini and Caporaso (Adv Math 240:1–23, 2013); Baker and Norine (Adv Math 215(2):766–788, 2007); Gathmann and Kerber (Math Z 259(1):217–230, 2008) and Mikhalkin and Zharkov (Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics 203–231, 2007), providing a common generalization of all of these results. For a complete nonsingular curve \(X\) defined over a non-Archimedean field \(\mathbb {K}\), together with a strongly semistable model \(\mathfrak {X}\) for \(X\) over the valuation ring \(R\) of \(\mathbb {K}\), we define a corresponding metrized complex \(\mathfrak {C}\mathfrak {X}\) of curves over the residue field \(\kappa \) of \(\mathbb {K}\) and a canonical specialization map \(\tau ^{\mathfrak {C}\mathfrak {X}}_*\) from divisors on \(X\) to divisors on \(\mathfrak {C}\mathfrak {X}\) which preserves degrees and linear equivalence. We then establish generalizations of the specialization lemma from Baker (Algebra Number Theory 2(6):613–653, 2008) and its weighted graph analogue from Amini and Caporaso (Adv Math 240:1–23, 2013), showing that the rank of a divisor cannot go down under specialization from \(X\) to \(\mathfrak {C}\mathfrak {X}\). As an application, we establish a concrete link between specialization of divisors from curves to metrized complexes and the theory of limit linear series due to Eisenbud and Harris (Invent Math 85:337–371, 1986). Using this link, we formulate a generalization of the notion of limit linear series to curves which are not necessarily of compact type and prove, among other things, that any degeneration of a \(\mathfrak {g}^r_d\) in a regular family of semistable curves is a limit \(\mathfrak {g}^r_d\) on the special fiber.



The authors would like to thank Vladimir Berkovich, Lucia Caporaso, Ethan Cotterill, Eric Katz, Johannes Nicaise, Joe Rabinoff, Frank-Olaf Schreyer, David Zureick-Brown, and the referees for helpful discussions and remarks. The second author was supported in part by NSF grant DMS-0901487.


  1. 1.
    Amini, O.: Equidistribution of Weierstrass points on curves over non-Archimedean fields, in preparationGoogle Scholar
  2. 2.
    Amini, O., Baker, M.: Limit linear series for a generic chain of genus one curves, in preparationGoogle Scholar
  3. 3.
    Amini, O.: Reduced divisors and embeddings of tropical curves. Trans. Am. Math. Soc. 365(9), 4851–4880 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Preprint arxiv:1303.4812
  5. 5.
    Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms II: Tropical curves and metrized complexes. Preprint arxiv:1404.3390
  6. 6.
    Amini, O., Caporaso, L.: Riemann-Roch theory for weighted graphs and tropical curves. Adv. Math. 240, 1–23 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory 2(6), 613–653 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Baker, M., Norine, S.: Riemann–Roch and Abel–Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Baker, M., Payne, S., Rabinoff, J.: Non-Archimedean geometry, tropicalization, and metrics on curves. Preprint arXiv:1104.0320v1
  10. 10.
    Baker, M., Shokrieh, F.: Chip-firing games, potential theory on graphs, and spanning trees. J. Comb. Theory Series A 120(1), 164–182 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Berkovich, V.G.: Spectral theory and analytic geometry over non-Archimedean fields. In: Proceedings of Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence (1990)Google Scholar
  12. 12.
    Bigas, M.T.I.: Brill-Noether theory for stable vector bundles. Duke Math. J. 62(2), 385–400 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Caporaso, L.: Linear series on semistable curves. Int. Math. Res. Not. 13, 2921–2969 (2011)Google Scholar
  14. 14.
    Caporaso, L.: Gonality of algebraic curves and graphs. In: Frühbis-Krüger A, Kloosterman RN, Schütt M (eds) Algebraic and Complex Geometry, Springer Proceedings in Mathematics & Statistics, vol 71. Springer, p 319 (2014)Google Scholar
  15. 15.
    Cartwright, D.: Lifting rank 2 tropical divisors. Preprint
  16. 16.
    Chinburg, T., Rumely, R.: The capacity pairing. J. für die reine und angewandte Mathematik 434, 1–44 (1993)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Cools, F., Draisma, J., Payne, S., Robeva, E.: A tropical proof of the Brill–Noether theorem. Adv. Math. 230(2), 759–776 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Coleman, R.F.: Effective Chabauty. Duke Math. J. 52(3), 765–770 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Publications Mathématiques de l’IHES 36(1), 75–109 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math. 85, 337–371 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Eisenbud, D., Harris, J.: The Kodaira dimension of the moduli space of curves of genus \({\ge }23\). Invent. Math. 90(2), 359–387 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Eisenbud, D., Harris, J.: Existence, decomposition, and limits of certain Weierstrass points. Invent. Math. 87(3), 495–515 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Eisenbud, D., Harris, J.: The monodromy of Weierstrass points. Invent. Math. 90(2), 333–341 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Esteves, E.: Linear systems and ramification points on reducible nodal curves. Mathematica Contemporanea 14, 21–35 (1998)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Esteves, E., Medeiros, N.: Limit canonical systems on curves with two components. Invent. Math. 149(2), 267–338 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Math. Z. 259(1), 217–230 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, Berlin (1998)Google Scholar
  28. 28.
    Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. Math. 67, 23–88 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Hartshorne, R.: Algebraic geometry. In: Proceedings of Springer Graduate Texts in Mathematics, p. 52 (1977)Google Scholar
  30. 30.
    Hladký, J., Kràl’, D., Norine, S.: Rank of divisors on tropical curves. J. Comb. Theory. Series B. 120(7), 1521–1538 (2013)CrossRefGoogle Scholar
  31. 31.
    Katz, E., Zureick-Brown, D.: The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions. Compositio Math. 149(11), 1818–1838 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Lorenzini, D.J., Tucker, T.J.: Thue equations and the method of Chabauty–Coleman. Invent. Math. 148(1), 47–77 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Lim, C.M., Payne, S., Potashnik, N.: A note on Brill–Noether thoery and rank determining sets for metric graphs. Int. Math. Res. Not. 23, 5484–5504 (2012)MathSciNetGoogle Scholar
  34. 34.
    Luo, Y.: Rank-determining sets of metric graphs. J. Comb. Theory. Series A. 118(6), 1775–1793 (2011)CrossRefzbMATHGoogle Scholar
  35. 35.
    McCallum, W., Poonen, B.: The method of Chabauty and Coleman, June 14, 2010. Preprint, to appear in Panoramas et Synthèses, Société Math. de France
  36. 36.
    Mikhalkin, G., Zharkov, I.: Tropical curves, their Jacobians and Theta functions. Contemporary Mathematics. In: Proceedings of the International Conference on Curves and Abelian Varieties in Honor of Roy Smith’s 65th Birthday, vol. 465, pp. 203–231 (2007)Google Scholar
  37. 37.
    Neeman, A.: The distribution of Weierstrass points on a compact Riemann surface. Ann. Math. 120, 317–328 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Osserman, B.: A limit linear series moduli scheme (Un schéma de modules de séries linéaires limites). Ann. Inst. Fourier 56(4), 1165–1205 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Osserman, B.: Linked Grassmannians and crude limit linear series. Int. Math. Res. Not. 25, 1–27 (2006)MathSciNetGoogle Scholar
  40. 40.
    Parker, B.: Exploded manifolds. Adv. Math. 229(6), 3256–3319 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Payne, S.: Fibers of tropicalization. Math. Zeit. 262, 301–311 (2009)CrossRefzbMATHGoogle Scholar
  42. 42.
    Ran, Z.: Modifications of Hodge bundles and enumerative geometry I: the stable hyperelliptic locus. Preprint arXiv:1011.0406
  43. 43.
    Stoll, M.: Independence of rational points on twists of a given curve. Compositio Math. 142(5), 1201–1214 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Temkin, M.: On local properties of non-Archimedean analytic spaces. Math. Annalen 318, 585–607 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Zhang, S.-W.: Admissible pairing on a curve. Invent. Math. 112(1), 171–193 (1993)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.CNRS-DMAÉcole Normale SupérieureParisFrance
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations