# Linear series on metrized complexes of algebraic curves

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## Abstract

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.

## Notes

### Acknowledgments

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.

## References

- 1.Amini, O.: Equidistribution of Weierstrass points on curves over non-Archimedean fields, in preparationGoogle Scholar
- 2.Amini, O., Baker, M.: Limit linear series for a generic chain of genus one curves, in preparationGoogle Scholar
- 3.Amini, O.: Reduced divisors and embeddings of tropical curves. Trans. Am. Math. Soc.
**365**(9), 4851–4880 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 4.Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta. Preprint arxiv:1303.4812
- 5.Amini, O., Baker, M., Brugallé, E., Rabinoff, J.: Lifting harmonic morphisms II: Tropical curves and metrized complexes. Preprint arxiv:1404.3390
- 6.Amini, O., Caporaso, L.: Riemann-Roch theory for weighted graphs and tropical curves. Adv. Math.
**240**, 1–23 (2013)CrossRefzbMATHMathSciNetGoogle Scholar - 7.Baker, M.: Specialization of linear systems from curves to graphs. Algebra Number Theory
**2**(6), 613–653 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Baker, M., Payne, S., Rabinoff, J.: Non-Archimedean geometry, tropicalization, and metrics on curves. Preprint arXiv:1104.0320v1
- 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.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.Bigas, M.T.I.: Brill-Noether theory for stable vector bundles. Duke Math. J.
**62**(2), 385–400 (1991)CrossRefzbMATHMathSciNetGoogle Scholar - 13.Caporaso, L.: Linear series on semistable curves. Int. Math. Res. Not.
**13**, 2921–2969 (2011)Google Scholar - 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.Cartwright, D.: Lifting rank 2 tropical divisors. Preprint http://users.math.yale.edu/dc597/lifting.pdf/
- 16.Chinburg, T., Rumely, R.: The capacity pairing. J. für die reine und angewandte Mathematik
**434**, 1–44 (1993)zbMATHMathSciNetGoogle Scholar - 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.Coleman, R.F.: Effective Chabauty. Duke Math. J.
**52**(3), 765–770 (1985)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Eisenbud, D., Harris, J.: Limit linear series: basic theory. Invent. Math.
**85**, 337–371 (1986)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Eisenbud, D., Harris, J.: Existence, decomposition, and limits of certain Weierstrass points. Invent. Math.
**87**(3), 495–515 (1987)CrossRefzbMATHMathSciNetGoogle Scholar - 23.Eisenbud, D., Harris, J.: The monodromy of Weierstrass points. Invent. Math.
**90**(2), 333–341 (1987)CrossRefzbMATHMathSciNetGoogle Scholar - 24.Esteves, E.: Linear systems and ramification points on reducible nodal curves. Mathematica Contemporanea
**14**, 21–35 (1998)zbMATHMathSciNetGoogle Scholar - 25.Esteves, E., Medeiros, N.: Limit canonical systems on curves with two components. Invent. Math.
**149**(2), 267–338 (2002)CrossRefzbMATHMathSciNetGoogle Scholar - 26.Gathmann, A., Kerber, M.: A Riemann–Roch theorem in tropical geometry. Math. Z.
**259**(1), 217–230 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 27.Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, Berlin (1998)Google Scholar
- 28.Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. Invent. Math.
**67**, 23–88 (1982)CrossRefzbMATHMathSciNetGoogle Scholar - 29.Hartshorne, R.: Algebraic geometry. In: Proceedings of Springer Graduate Texts in Mathematics, p. 52 (1977)Google Scholar
- 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.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.Lorenzini, D.J., Tucker, T.J.: Thue equations and the method of Chabauty–Coleman. Invent. Math.
**148**(1), 47–77 (2002)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Luo, Y.: Rank-determining sets of metric graphs. J. Comb. Theory. Series A.
**118**(6), 1775–1793 (2011)CrossRefzbMATHGoogle Scholar - 35.McCallum, W., Poonen, B.: The method of Chabauty and Coleman, June 14, 2010. Preprint http://www-math.mit.edu/poonen/papers/chabauty.pdf, to appear in Panoramas et Synthèses, Société Math. de France
- 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.Neeman, A.: The distribution of Weierstrass points on a compact Riemann surface. Ann. Math.
**120**, 317–328 (1984)CrossRefzbMATHMathSciNetGoogle Scholar - 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.Osserman, B.: Linked Grassmannians and crude limit linear series. Int. Math. Res. Not.
**25**, 1–27 (2006)MathSciNetGoogle Scholar - 40.Parker, B.: Exploded manifolds. Adv. Math.
**229**(6), 3256–3319 (2012)CrossRefzbMATHMathSciNetGoogle Scholar - 41.Payne, S.: Fibers of tropicalization. Math. Zeit.
**262**, 301–311 (2009)CrossRefzbMATHGoogle Scholar - 42.Ran, Z.: Modifications of Hodge bundles and enumerative geometry I: the stable hyperelliptic locus. Preprint arXiv:1011.0406
- 43.Stoll, M.: Independence of rational points on twists of a given curve. Compositio Math.
**142**(5), 1201–1214 (2006)CrossRefzbMATHMathSciNetGoogle Scholar - 44.Temkin, M.: On local properties of non-Archimedean analytic spaces. Math. Annalen
**318**, 585–607 (2000)CrossRefzbMATHMathSciNetGoogle Scholar - 45.Zhang, S.-W.: Admissible pairing on a curve. Invent. Math.
**112**(1), 171–193 (1993)CrossRefzbMATHMathSciNetGoogle Scholar