Tropical fans and the moduli spaces of tropical curves

By Andreas Gathmann, Michael Kerber, Hannah Markwig. Compositio Mathematica, 2007

125 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Oncology

View Paper: https://www.semanticscholar.org/paper/73068d143bb8431cd207077a94e3d559f10debee

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Citations in the corpus

On the tropical Torelli map (2009, background): “Though, the moduli space of n-pointed tropical rational curves M tr 0,n was constructed by different authors (see [56, 42,23,30]).”
Combinatorics of the tropical Torelli map (2010, background): “” Contrast this with the situation of moduli space M0,n of tropical rational curves with n marked points, constructed and studied in [Speyer and Sturmfels 2004; Mikhalkin 2007; Gathmann et al. 2009].”
Algebraic and tropical curves: comparing their moduli spaces (2011, background): “The case g = 0, which has been extensively studied (see for example [22], [23], [14], [16]), is an exception, because of the absence of automorphisms.”
A MODULI STACK OF TROPICAL CURVES (2017, unknown): “oints, and variants of it, are some of the most-studied objects in tropical geometry (see e.g. [Cap13, CMV13, Cap16, Cha16] for survey papers). In particular, we refer the reader to [Mik06], [Mik07], [GKM09], and [GM08] for the case of genus g = 0, as well as to [CV10], [BMV11], [Cha12], and [Viv13]”