Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 4

Special Issue: In Honor of Prof. Gert-Martin Greuel’s 75th Birthday

Guest Editors: Igor Burban, Stanislaw Janeczko, Gerhard Pfister, Stephen S.T. Yau, and Huaiqing Zuo

On refined count of rational tropical curves

Pages: 1027 – 1052

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n4.a5


Eugenii Shustin (School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel)


We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked four-valent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants.


rational tropical curve, plane tropical curve, spacial tropical curve, tropical enumerative geometry, refined enumerative invariants

2010 Mathematics Subject Classification

Primary 14T05. Secondary 14N10.

The author has been supported by the Israel Science Foundation grants no. 176/15 and 501/18, and by The Bauer-Neuman Chair in Real and Complex Geometry.

Received 8 December 2018

Accepted 2 July 2019

Published 13 November 2020