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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Reduced divisors and embeddings of tropical curves


Author: Omid Amini
Journal: Trans. Amer. Math. Soc. 365 (2013), 4851-4880
MSC (2010): Primary 14T05; Secondary 14C20, 14A10, 05C10
Published electronically: April 2, 2013
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Abstract: Given a divisor $ D$ on a tropical curve $ \Gamma $, we show that reduced divisors define an integral affine map from the tropical curve to the complete linear system $ \vert D\vert$. 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 reduced-divisor 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 reduced-divisor 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 rank-determining sets of points. We also discuss the classical analogue of the (tropical) reduced-divisor map: For a smooth projective curve $ C$ and a divisor $ D$ of non-negative rank on $ C$, reduced divisors equivalent to $ D$ define a morphism from $ C$ to the complete linear system $ \vert D\vert$, which is described in terms of Wronskians.


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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/S0002-9947-2013-05789-3
PII: S 0002-9947(2013)05789-3
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.