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Tropical fans and the moduli spaces of tropical curves

Part of: Projective and enumerative geometry Real and complex geometry

Published online by Cambridge University Press:  01 January 2009

Andreas Gathmann ,
Michael Kerber  and
Hannah Markwig
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)
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Abstract

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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.

Keywords

tropical geometrytropical curvesenumerative geometry

MSC classification

Secondary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 51M20: Polyhedra and polytopes; regular figures, division of spaces 14N10: Enumerative problems (combinatorial problems)
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 173 - 195
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), 733–767.CrossRefGoogle Scholar
[2]Bogart, T., Jensen, A., Speyer, D., Sturmfels, B. and Thomas, R., Computing tropical varieties, J. Symbolic Comput. 42 (2007), 54–73.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), 845–868.Google Scholar
[4]Gathmann, A. and Markwig, H., Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537–560.CrossRefGoogle Scholar
[5]Mikhalkin, G., Enumerative tropical geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313–377.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), 1–51.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), 389–411.CrossRefGoogle Scholar
[11]Sturmfels, B. and Tevelev, J., Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), 543–562.Google Scholar