The numbers of tropical plane curves through points in general position

By Andreas Gathmann, Hannah Markwig. 2005

87 citations recorded by the source corpus.

Abstract We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral vectors this statement has been known for a while now, but the only known proof was indirect with the help of Mikhalkin's Correspondence Theorem that translates this question into the well-known fact that the numbers of complex curves in a toric surface through some given points do not depend on the position of the points. This paper presents a direct proof entirely within tropical geometry that is in addition applicable to arbitrary degree of the curves.

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

View Paper: https://www.semanticscholar.org/paper/844fc63b0bf931f4c2e3de30f266ff3aad24cc24

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In Computational Mechanics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Logarithmic Gromov–Witten theory with expansions (2019, background): “When the target is a toric surface degeneration, this simplification is implicit throughout the tropical literature, see for instance [27].”
On Block-Goettsche multiplicities for planar tropical curves (2012, background): “9 of [7] concerns the dimensions of the moduli spacesMg,∆, of tropical curves h : C → R(2) of genus g′ ≤ g and degree ∆ with k marked points y1, .”
The tropical vertex (2009, result): “The result follows from standard tropical arguments — for example the method of [6] suffices.”
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces (2006, background): “In the case α = (0), the moduli space M∆,α,β,g coincides with a moduli space introduced by Gathmann and Markwig [2] (in [2] this space is denoted by Mg,∆, where ∆ = ∆(0),β).”