Addition in Jacobians of tropical hyperelliptic curves
type: Item
instance of: scholarly article
zbMATH Open document ID: 1273.14134
author: Atsushi Nobe
publication date: 2012-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1111.6657
Mathematics Subject Classification ID: 14T05
Mathematics Subject Classification ID: 14H70
zbMATH DE Number: 6088931
zbMATH Keywords: tropical hyperelliptic curve
zbMATH Keywords: Jacobian
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: rational function
MaRDI profile type: Publication
arXiv ID: 1111.6657
arXiv classification: nlin.SI
arXiv classification: math.AG
title: Addition in Jacobians of tropical hyperelliptic curves
published in: RIMS Kôkyûroku Bessatsu
review text: The author shows that there is a surjection (namely, the Abel--Jacobi map associated to a base point) from the set of effective divisors of degree \(g\) on a tropical curve of genus \(g\) to the curve's Jacobian. This follows in a few lines from the tropical Riemann--Roch theorem of \textit{G. B. Mikhalkin} and \textit{I. Zharkov} [Contemporary Mathematics 465, 203--230 (2008; Zbl 1152.14028)] and \textit{A. Gathmann} and \textit{M. Kerber} [Math. Z. 259, No. 1, 217--230 (2008; Zbl 1187.14066)].NEWLINENEWLINEThe author then cites a theorem from [\textit{R. Inoue} and \textit{T. Takenawa}, Int. Math. Res. Not. 2008, Article ID rnn019, 27 p. (2008; Zbl 1155.37012)] claiming that the Abel--Jacobi map restricts to a bijection between a suitable open subset of the curve, and the Jacobian. In the cited paper, this is only proven for \(g\leq 3\), although the authors claim to have a complete proof. The reviewer is uncertain as to whether a general proof has been published.NEWLINENEWLINEThe greater part of the paper under review is then devoted to a geometric description of the additive group structure pulled back from the Jacobian to the open subset of the set of effective divisors. The addition is described in terms of intersection with an auxiliary tropical curve of degree \(3\lfloor g/2\rfloor\).
reviewed by: Péter E. Frenkel
description: scientific article; zbMATH DE number 6088931