Compositio Mathematica

Research Article

Tropical fans and the moduli spaces of tropical curves

Andreas Gathmanna1, Michael Kerbera2 and Hannah Markwiga3

a1 Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (email: andreas@mathematik.uni-kl.de)

a2 Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (email: mkerber@mathematik.uni-kl.de)

a3 Institute for Mathematics and its Applications (IMA), University of Minnesota, Lind Hall 400, 207 Church Street SE, Minneapolis, MN 55455, USA (email: markwig@ima.umn.edu)

Abstract

We give a rigorous definition of tropical fans (the ‘local building blocks for tropical varieties’) and their morphisms. For a morphism of tropical fans of the same dimension we show that the number of inverse images (counted with suitable tropical multiplicities) of a point in the target does not depend on the chosen point; a statement that can be viewed as one of the important first steps of tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some r) together with the evaluation and forgetful morphisms. Using our results this gives new, easy and unified proofs of various tropical independence statements, e.g. of the fact that the numbers of rational tropical curves (in any r) through given points are independent of the points.

(Received November 10 2007)

(Accepted August 05 2008)

2000 Mathematics Subject Classification

  • 14N35;
  • 51M20 (primary);
  • 14N10 (secondary)

Keywords

  • tropical geometry;
  • tropical curves;
  • enumerative geometry

Footnotes

The second author would like to thank the Institute for Mathematics and its Applications (IMA) in Minneapolis for their hospitality. The third author would like to thank Ionut Ciocan-Fontanine, Diane Maclagan and Grisha Mikhalkin for many helpful discussions.