Tropical fans and the moduli spaces of tropical curves

Gathmann, Andreas; Kerber, Michael; Markwig, Hannah

doi:10.1112/s0010437x08003837

Cited by 80 publications

(183 citation statements)

References 9 publications

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“…The tropical analogue M 0,n of the space of stable n-marked curves is the space of trees, or (a quotient of) the tropical Grassmanian [4,12,14]. Thus an abstract tropical curve is just a tree with n marked ends and whose bounded edges e are equipped with a length l(e) ∈ R >0 .…”

Section: Defining the Invariantsmentioning

confidence: 99%

“…In Sect. 4, we show that every tropical curve in an intersection product of Psi-classes, point and line evaluations and the pullback of a point with a large coordinate in M 0,4 under the forgetful map contains a contracted bounded edge. Thus the tropical curve can be interpreted as a reducible curve by cutting it along this contracted bounded edge.…”

Section: Introductionmentioning

confidence: 99%

“…maps on the latter ones [4,12,14]. Moreover, [12] already defines Psi-classes on the space of abstract tropical curves M 0,n and [7] deals with their intersections.…”

Section: Introductionmentioning

confidence: 99%

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Tropical descendant Gromov–Witten invariants

2009

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Abstract. We define tropical Psi-classes on M 0,n (R 2 , d) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psiand evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin's lattice path algorithm and counts rational plane tropical curves satisfying certain Psi-and evaluation conditions.

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Section: Defining the Invariantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Tropical descendant Gromov–Witten invariants

2009

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“…For a more detailed account, see [10,22,23]. The thick red lines indicate the flat, which consists of two complete subgraphs.…”

Section: Tropical Moduli Spaces Of Rational Weighted Curvesmentioning

confidence: 99%

“…The cone complex M trop 0,n can be embedded into a vector space, and given the structure of a balanced fan, by assigning weight 1 to each top-dimensional cone. In [10,22], M trop 0,n is discussed in analogy with the classical moduli space, with the goal of understanding its tropical intersection theory. However, the connection between the algebraic and tropical moduli spaces runs much deeper.…”

Section: Context and Motivationmentioning

confidence: 99%

Moduli Spaces of Rational Weighted Stable Curves and Tropical Geometry

Cavalieri

Hampe

Markwig

et al. 2016

Forum of Mathematics, Sigma

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We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector w of weights, the moduli space of tropical w-stable curves can be given the structure of a balanced fan if and only if w has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

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Enumerative Aspects of the Gross-Siebert Program

Garrel¹,

Peter²,

Ruddat³

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

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We present enumerative aspects of the Gross-Siebert program in this introductory survey. After sketching the program's main themes and goals, we review the basic definitions and results of logarithmic and tropical geometry. We give examples and a proof for counting algebraic curves via tropical curves. To illustrate an application of tropical geometry and the Gross-Siebert program to mirror symmetry, we discuss the mirror symmetry of the projective plane.Michel van Garrel KIAS,discretizing Hitchin's Legendre duality.Enumerative aspects of the Gross-Siebert program 3 Kontsevich and Soibelman [32] demonstrated how one could reconstruct a K3 surface from an affine structure with singularities on S 2 . Using logarithmic geometry, Gross and Siebert were able to solve the reconstruction problem [20] in any dimension, obtaining a degenerating family of Calabi-Yau manifolds X → D over a holomorphic disk from the information of (B, P, ϕ) and a log structure. Furthermore, this family is parametrized by a canonical coordinate (in the usual sense in mirror symmetry). The construction features wall-crossings and scatterings, structures that encode enumerative information linking symplectic with complex geometry via tropical geometry. As will be hinted at in this exposition, Gromov-Witten theory [21] can also be incorporated in this framework. Toric conventionsWe assume familiarity with toric geometry. The interested reader is referred to the excellent exposition of Fulton [10]. As the following story is closely tied to toric geometry, it is convenient to begin by making a few conventions regarding notation.SetFor n ∈ N, set n, m to be the evaluation of n on m. Set a toric fan Σ in M R . Let Σ [n] signify the set of n dimensional cones of Σ . Let X Σ be the toric variety defined by Σ .Denote by T Σ the free abelian group generated by Σ [1] . For ρ ∈ Σ [1] , denote by v ρ the corresponding generator in T Σ . We will need the mapwhereρ is the integral vector generating ρ, that is ρ ∩ M = Z ≥0ρ .

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Tropical fans and the moduli spaces of tropical curves

Cited by 80 publications

References 9 publications

Tropical descendant Gromov–Witten invariants

Tropical descendant Gromov–Witten invariants

Moduli Spaces of Rational Weighted Stable Curves and Tropical Geometry

Enumerative Aspects of the Gross-Siebert Program

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