Addition in Jacobians of tropical hyperelliptic curves
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 \(3\lfloor g/2\rfloor\).
scientific article; zbMATH DE number 6088931