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HYPERELLIPTIC GRAPHS AND METRIZED COMPLEXES

Part of: Cycles and subschemes Curves

Published online by Cambridge University Press:  30 August 2017

YOAV LEN
YOAV LEN*
Affiliation:
Department of Combinatorics & Optimization, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada; [email protected]

Abstract

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We prove a version of Clifford’s theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree $2r$ and rank $r$ (for $0<r<g-1$) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens’ theorem for metric graphs.

MSC classification

Primary: 14C20: Divisors, linear systems, invertible sheaves 14H51: Special divisors (gonality, Brill-Noether theory) 14H40: Jacobians, Prym varieties 14T05: Tropical geometry
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 5 , 2017 , e20
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2017

References

Amini, O. and Baker, M., ‘Linear series on metrized complexes of algebraic curves’, Math. Ann. 362(1–2) (2015), 55–106.CrossRefGoogle Scholar
Amini, O., Baker, M., Brugallé, E. and Rabinoff, J., ‘Lifting harmonic morphisms II: tropical curves and metrized complexes’, Algebra Number Theory 9(2) (2015), 267–315.Google Scholar
Amini, O. and Caporaso, L., ‘Riemann–Roch theory for weighted graphs and tropical curves’, Adv. Math. 240 (2013), 1–23.CrossRefGoogle Scholar
Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J., Geometry of Algebraic Curves, Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences, 267] (Springer, New York, 1985).CrossRefGoogle Scholar
Baker, M., ‘Specialization of linear systems from curves to graphs’, Algebra Number Theory 2(6) (2008), 613–653.Google Scholar
Baker, M. and Faber, X., ‘Metric properties of the tropical Abel–Jacobi map’, J. Algebraic Combin. 33(3) (2011), 349–381.Google Scholar
Baker, M. and Jensen, D., Degeneration of Linear Series from the Tropical Point of View and Applications (Springer, Cham, 2016), 365–433.Google Scholar
Baker, M. and Norine, S., ‘Riemann–Roch and Abel–Jacobi theory on a finite graph’, Adv. Math. 215(2) (2007), 766–788.CrossRefGoogle Scholar
Chan, M., ‘Tropical hyperelliptic curves’, J. Algebraic Combin. 37(2) (2013), 331–359.Google Scholar
Coppens, M., ‘Clifford’s theorem for graphs’, Preprint, 2013, arXiv:1304.6101.Google Scholar
Gathmann, A. and Kerber, M., ‘A Riemann–Roch theorem in tropical geometry’, Math. Z. 259(1) (2008), 217–230.Google Scholar
Griffiths, P. and Harris, J., ‘On the variety of special linear systems on a general algebraic curve’, Duke Math. J. 47(1) (1980), 233–272.Google Scholar
Haase, C., Musiker, G. and Yu, J., ‘Linear systems on tropical curves’, Math. Z. 270(3–4) (2012), 1111–1140.Google Scholar
Katz, E., Rabinoff, J. and Zureick-Brown, D., ‘Uniform bounds for the number of rational points on curves of small mordell–weil rank’, Duke Math. J. 165(16) (2016), 3189–3240.CrossRefGoogle Scholar
Kawaguchi, S. and Yamaki, K., ‘Rank of divisors on hyperelliptic curves and graphs under specialization’, Int. Math. Res. Not. IMRN 2015(12) (2015), 4121–4176.Google Scholar
Kempf, G., ‘Schubert methods with an application to algebraic curves’, Publ. Math. Centrum (1971).Google Scholar
Kleiman, L. and Laksov, D., ‘On the existence of special divisors’, Amer. J. Math. 94(2) (1972), 431–436.Google Scholar
Len, Y., ‘The Brill–Noether rank of a tropical curve’, J. Algebraic Combin. 40(3) (2014), 841–860.Google Scholar
Lim, C. M., Payne, S. and Potashnik, N., ‘A note on Brill–Noether theory and rank-determining sets for metric graphs’, Int. Math. Res. Not. IMRN 2012(23) (2012), 5484–5504.CrossRefGoogle Scholar
Mikhalkin, G. and Zharkov, I., ‘Tropical curves, their Jacobians and theta functions’, inCurves and Abelian Varieties, Contemporary Mathematics, 465 (American Mathematical Society, Providence, RI, 2008), 203–230.Google Scholar