Kontsevich's formula and the WDVV equations in tropical geometry

By Andreas Gathmann, Hannah Markwig. 2005

96 citations recorded by the source corpus.

Abstract Using Gromov–Witten theory the numbers of complex plane rational curves of degree d through 3 d − 1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d2aecf6f5cee97bec4965b045ebedeff7b6e9f10

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Statistical and Nonlinear Physics

Citations in the corpus

Moduli spaces of rational tropical curves (2007, methodology): “us the image would have to split as a union of several tropical curves in the target. Such decompositions were used by Gathmann and Markwig in their deduction of the tropical WDVV equation in R2, see [1]. 3. Tropical ψ-classes Note that we do have the forgetting maps ft j : M0,n+1 → M0,n 10 GRIGORYM”
A MODULI STACK OF TROPICAL CURVES (2017, unknown): “riants 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] for the high”
Algebraic and tropical curves: comparing their moduli spaces (2011, background): “The case g = 0, which has been extensively studied (see for example [21], [22], [13], [15]), is an exception, because of the absence of automorphisms.”
A Riemann–Roch theorem in tropical geometry (2006, background): “Note that these maps are morphisms of polyhedral complexes in the sense of [ GM ] (i.e.”
Tropical Geometry and its applications (2006, unknown): “ll-backs and then take their intersection number in Mβ g,k (X). Many reasonings in the Gromov-Witten theory can be literally repeated in this tropical set-up. A good example is the WDVV-relation, cf. [7]. As in the classical case the stable curves in Mβ 0,k(X)rM β ,k (X) must consist of several comp”
Tropical geometry and correspondence theorems via toric stacks (2010, background): “In [7], Gathmann and Markwig defined stabilization for tropi cal curves without one-valent finite vertices.”
On the tropical Torelli map (2009, background): “For instance, this program has been carried out successfully for many problems of real and complex enumerative geometry, see for example [32,40,50,24,25,11,33,7,21].”
Tropical fans and the moduli spaces of tropical curves (2007, methodology): “DUCTION Tropical geometry has been used recently for many applications in enumerative geometry, both to reprove previously known results (e.g. the Caporaso-Harris formula [GM2] and Kontsevich’sformula[GM3] forplanecurves)andto ﬁndnew m ethodsforcountingcurves (e.g. Mikhalkin’s lattice path algorithm”