Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-29T09:23:23.047Z Has data issue: false hasContentIssue false

Tropical fans and the moduli spaces of tropical curves

Published online by Cambridge University Press:  01 January 2009

Andreas Gathmann
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (email: andreas@mathematik.uni-kl.de)
Michael Kerber
Affiliation:
Fachbereich Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany (email: mkerber@mathematik.uni-kl.de)
Hannah Markwig
Affiliation:
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)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 ℝr) 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 ℝr) through given points are independent of the points.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Billera, L., Holmes, S. and Vogtmann, K., Geometry of the space of phylogenetic trees, Adv. Appl. Math. 27 (2001), 733767.CrossRefGoogle Scholar
[2]Bogart, T., Jensen, A., Speyer, D., Sturmfels, B. and Thomas, R., Computing tropical varieties, J. Symbolic Comput. 42 (2007), 5473.CrossRefGoogle Scholar
[3]Gathmann, A. and Markwig, H., The Caporaso–Harris formula and plane relative Gromov–Witten invariants in tropical geometry, Math. Ann. 338 (2007), 845868.Google Scholar
[4]Gathmann, A. and Markwig, H., Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537560.CrossRefGoogle Scholar
[5]Mikhalkin, G., Enumerative tropical geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313377.CrossRefGoogle Scholar
[6]Mikhalkin, G., Tropical geometry and its applications, in Proc. int. conf. on mathematics, Madrid, Spain, 2006, 827–852 (math.AG/0601041).Google Scholar
[7]Mikhalkin, G., Moduli spaces of rational tropical curves, in Proc. Gökova geometry-topology conf. (GGT), Gökova, 2007, 39–51 (arXiv/0704.0839).Google Scholar
[8]Nishinou, T. and Siebert, B., Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 151.CrossRefGoogle Scholar
[9]Speyer, D., Tropical geometry, PhD thesis, University of California, Berkeley, CA (2005).Google Scholar
[10]Speyer, D. and Sturmfels, B., The tropical Grassmannian, Adv. Geom. 4 (2004), 389411.CrossRefGoogle Scholar
[11]Sturmfels, B. and Tevelev, J., Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), 543562.Google Scholar