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Tropical fans and the moduli spaces of tropical curves

  • Andreas Gathmann (a1), Michael Kerber (a2) and Hannah Markwig (a3)
Abstract
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 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.

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[1]L. Billera , S. Holmes and K. Vogtmann , Geometry of the space of phylogenetic trees, Adv. Appl. Math. 27 (2001), 733767.

[2]T. Bogart , A. Jensen , D. Speyer , B. Sturmfels and R. Thomas , Computing tropical varieties, J. Symbolic Comput. 42 (2007), 5473.

[3]A. Gathmann and H. Markwig , The Caporaso–Harris formula and plane relative Gromov–Witten invariants in tropical geometry, Math. Ann. 338 (2007), 845868.

[4]A. Gathmann and H. Markwig , Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008), 537560.

[5]G. Mikhalkin , Enumerative tropical geometry in ℝ2, J. Amer. Math. Soc. 18 (2005), 313377.

[8]T. Nishinou and B. Siebert , Toric degenerations of toric varieties and tropical curves, Duke Math. J. 135 (2006), 151.

[10]D. Speyer and B. Sturmfels , The tropical Grassmannian, Adv. Geom. 4 (2004), 389411.

[11]B. Sturmfels and J. Tevelev , Elimination theory for tropical varieties, Math. Res. Lett. 15 (2008), 543562.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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