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

DOI:
https://doi.org/10.1090/S0002-9947-2013-05789-3

Published electronically:
April 2, 2013

MathSciNet review:
3066772

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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 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 and a divisor of non-negative rank on , reduced divisors equivalent to define a morphism from to the complete linear system , 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:
https://doi.org/10.1090/S0002-9947-2013-05789-3

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.