Reduced divisors and embeddings of tropical curves
Author:
Omid Amini
Journal:
Trans. Amer. Math. Soc. 365 (2013), 48514880
MSC (2010):
Primary 14T05; Secondary 14C20, 14A10, 05C10
Published electronically:
April 2, 2013
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Abstract: Given a divisor on a tropical curve , we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system . This is done by providing an explicit description of the behavior of reduced divisors under infinitesimal modifications of the base point. We consider the cases where the reduceddivisor map defines an embedding of the curve into the linear system and, in this way, classify all the tropical curves with a very ample canonical divisor. As an application of the reduceddivisor map, we show the existence of Weierstrass points on tropical curves of genus at least two and present a simpler proof of a theorem of Luo on rankdetermining sets of points. We also discuss the classical analogue of the (tropical) reduceddivisor map: For a smooth projective curve and a divisor of nonnegative rank on , reduced divisors equivalent to define a morphism from to the complete linear system , which is described in terms of Wronskians.
 1.
Omid
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 1.
 O. Amini and M. Manjunath, RiemannRoch for sublattices of the root lattice , Electron. J. Combin. 17 (2010), no. 1, Research Paper 124, 50 pp. MR 2729373 (2012a:06009)
 2.
 M. Baker, Specialization of Linear Systems From Curves to Graphs. With an appendix by B. Conrad, Algebra Number Theory 2 (2008), no. 6, 613653. MR 2448666 (2010a:14012)
 3.
 M. Baker and S. Norine, RiemannRoch and AbelJacobi theory on a finite graph, Adv. Math. 215 (2007), no. 2, 766788. MR 2355607 (2008m:05167)
 4.
 A. Bostan and P. Dumas, Wronskians and linear independence, Amer. Math. Monthly 117 (2010), no. 8, 722727. MR 2732247 (2011g:15003)
 5.
 A. Gathmann and M. Kerber, RiemannRoch Theorem in Tropical Geometry, Math. Z. 259 (2008), no. 1, 217230. MR 2377750 (2009a:14014)
 6.
 C. Hasse, G. Musiker, and J. Yu, Linear Systems on Tropical Curves, Math. Z., to appear.
 7.
 J. Hladký, D. Kràl', and S. Norine, Rank of divisors on tropical curves, J. Combin. Theory Ser. A, to appear.
 8.
 Y. Luo, Rankdetermining sets of metric graphs, Combin. Theory Ser. A. 118 (2011), no. 6, 17751793. MR 2793609 (2012d:05122)
 9.
 G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties, Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, 2008, pp. 203230. MR 2457739 (2011c:14163)
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Additional Information
Omid Amini
Affiliation:
CNRS, Département de mathématiques et applications, École Normale Supérieure, 45 Rue d’Ulm, 75230 Paris Cedex 05, France
Email:
oamini@math.ens.fr
DOI:
http://dx.doi.org/10.1090/S000299472013057893
PII:
S 00029947(2013)057893
Received by editor(s):
March 9, 2011
Received by editor(s) in revised form:
November 22, 2011
Published electronically:
April 2, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
