Reduced divisors and embeddings of tropical curves
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- by Omid Amini PDF
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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 $|D|$. 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 $|D|$, which is described in terms of Wronskians.References
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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
- Received by editor(s): March 9, 2011
- Received by editor(s) in revised form: November 22, 2011
- Published electronically: April 2, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3066772