Tropical fans and the moduli spaces of tropical curves

Abstract

We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such 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 the beginning of a tropical intersection theory. As an application we consider the moduli spaces of rational tropical curves (both abstract and in some R^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^r) through given points are independent of the points.

minor corrections to match published version

References in corpus (2)

• Tropical Geometry and its applications
• Moduli spaces of rational tropical curves

Cited by in corpus (50)

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• On the tropical Torelli map
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• The diagonal of tropical matroid varieties and cycle intersections
• Skeletons of stable maps I: Rational curves in toric varieties
• Geometry of tropical moduli spaces and linkage of graphs
• Uniformizing Tropical Curves I: Genus Zero and One
• Tropical descendant Gromov-Witten invariants
• Psi-floor diagrams and a Caporaso-Harris type recursion
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• A-Tint: A polymake extension for algorithmic tropical intersection theory
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• Correspondence Theorems via Tropicalizations of Moduli Spaces
• Refined descendant invariants of toric surfaces
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• Tropical rational equivalence on R^r
• Generic Tropical Varieties
• Tropicalized quartics and canonical embeddings for tropical curves of genus 3
• Tropical Intersection Theory from Toric Varieties
• Universal families of rational tropical curves
• Tropicalization of the moduli space of stable maps
• Towards a tropical Hodge bundle
• Tropical geometry of genus two curves
• Tropical covers of curves and their moduli spaces
• A tropical intersection product in matroidal fans
• On refined count of rational tropical curves
• Combinatorics of the tropical Torelli map
• Algebraic Properties of Generic Tropical Varieties
• Tropical cycles and Chow polytopes
• Tropical enumerative invariants of F_0 and F_2
• Homology of tropical fans
• Chern classes of tropical vector bundles
• Tropical $\psi$ classes
• Combinatorics of tropical Hurwitz cycles
• Cocycles on tropical varieties via piecewise polynomials
• Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves
• Tropical Moduli Space of Rational Graphically Stable Curves
• Tropical intersection theory on R^n
• Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
• Idempotent Semigroups and Tropical Algebraic Sets
• Planar tropical gravitational descendants with one tangency condition
• Tropical Theta Functions and Riemann-Roch Inequality for Tropical Abelian Surfaces
• Minkowski weights and the Grothendieck group of a toric variety
• Topological Field Theories and Harrison Homology
• Moduli Spaces of Rational Graphically Stable Curves
• Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles
• Tropical orbit spaces and the moduli spaces of elliptic tropical curves