Tropical curves, graph complexes, and top weight cohomology of M_g

Chan, Melody; Galatius, Soren; Payne, Sam

doi:10.48550/arxiv.1805.10186

Cited by 15 publications

(49 citation statements)

References 24 publications

(45 reference statements)

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“…

This article is a sequel to [CGP18]. We study a space ∆ g,n of genus g stable, n-marked tropical curves with total edge length 1.

…”

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confidence: 99%

Topology of moduli spaces of tropical curves with marked points

Chan,

Galatius,

Payne

2019

Preprint

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This article is a sequel to [CGP18]. We study a space ∆ g,n of genus g stable, n-marked tropical curves with total edge length 1. Its rational homology is identified both with top-weight cohomology of the complex moduli space M g,n and with the homology of a marked version of Kontsevich's graph complex, up to a shift in degrees.We prove a contractibility criterion that applies to various large subspaces of ∆ g,n . From this we derive a description of the homotopy type of ∆ 1,n , the top weight cohomology of M 1,n as an S n -representation, and additional calculations of H i (∆ g,n ; Q) for small (g, n). We also deduce a vanishing theorem for homology of marked graph complexes from vanishing of cohomology of M g,n in appropriate degrees, by relating both to ∆ g,n . We comment on stability phenomena, or lack thereof.Marked graph complexes. In [CGP18] we gave a cellular chain complex C * (X; Q) associated to any symmetric ∆-complex X, and showed that it computes the reduced rational homology of (the geometric realization of) X. In the case X = ∆ g,n , we are able to deduce that C * (∆ g,n ; Q) is quasi-isomorphic to the marked graph complex G (g,n) , using Theorem 1.1. We shall define G (g,n) precisely in Section 2. Briefly, it is generated by isomorphism classes of connected, loopless, n-marked stable graphs Γ together with the choice of one of the two possible orientations of R E (Γ) . The main difference between G (g,n) and the hairy graph complexes in the existing literature [CKV13, CKV15, KW Ž16, TW17] is that we consider graphs with n labeled markings, rather than n unlabeled hairs or half-edges. 1 We use the terminology "marked" rather than "hairy" for this reason. Kontsevich's graph complex G (g) [Kon93, Kon94] occurs as the special case n = 0.Theorem 1.4. For 2g − 2 + n > 0, there is a natural split injection of chain complexes1 We expect that there may be natural maps between the S n -coinvariants of our marked graph complex and some existing hairy graph complexes, but have been unable to find such precise relations.

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“…

This article is a sequel to [CGP18]. We study a space ∆ g,n of genus g stable, n-marked tropical curves with total edge length 1.

…”

mentioning

confidence: 99%

Topology of moduli spaces of tropical curves with marked points

Chan,

Galatius,

Payne

2019

Preprint

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“…Description of ∆ g,w as a functor. We will calculate Aut(∆ g,w ) in the category of symmetric ∆-complexes, as introduced by Chan, Galatius, and Payne [CGP18]. Put I for the category whose objects are the sets [p] for each p ≥ 0, and whose morphisms are all injections.…”

Section: An Isomorphism Of Pairsmentioning

confidence: 99%

“…A symmetric ∆-complex X : I op → Set should be thought of as a set of combinatorial gluing instructions for a topological space |X|. There is a geometric realization functor given by X → |X|; see [CGP18], [Kan], or [KLSY20] for a description of this functor.…”

Section: An Isomorphism Of Pairsmentioning

confidence: 99%

“…An important feature of the compactifaction M g,n ⊂ M g,n is that the boundary divisor ∂M g,n := M g,n M g,n is normal crossings. In [CGP18], Chan, Galatius, and Payne, following work of Harper [Har17] and Abramovich-Caporaso-Payne [ACP15], show how to construct the dual complex ∆(X , D) of a normal crossings divisor D on a Deligne-Mumford stack X . They study ∆(X , D) in the case where X = M g,n and D = ∂M g,n , showing that ∆(X , D) = ∆ g,n is identified with the link of the cone point in the moduli space M trop g,n of stable n-marked tropical curves of genus g. On the other hand, the complement of M g,n in Hassett's compactification M g,w is not in general normal crossings.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we are interested in the automorphism groups of the complexes ∆ g,w , taken in the category of symmetric ∆-complexes, as defined in [CGP18] and recalled in Section 2. Given a weight vector w, we can form an abstract simplicial complex K w with vertex set {1, .…”

Section: Introductionmentioning

confidence: 99%

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Automorphisms of tropical Hassett spaces

Freedman¹,

Hlavinka²,

Kannan³

2021

Preprint

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Given an integer g ≥ 0 and a weight vector w ∈ Q n ∩(0, 1] n satisfying 2g−2+ wi > 0, let ∆g,w denote the moduli space of n-marked, w-stable tropical curves of genus g and volume one. We calculate the automorphism group Aut(∆g,w) for g ≥ 1 and arbitrary w, and we calculate the group Aut(∆0,w) when w is heavy/light. In both of these cases, we show that Aut(∆g,w) ∼ = Aut(Kw), where Kw is the abstract simplicial complex on {1, . . . , n} whose faces are subsets with w-weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space ∆g,w may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space Mg,w. Following the work of Massarenti and Mella [MM17] on the biregular automorphism group Aut(Mg,w), we show that Aut(∆g,w) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves. Contents1. Introduction 1 2. Graphs and ∆ g,w 5 3. Calculation of Aut(∆ g,w ) for g ≥ 1 8 4. The genus 0 case 14 Appendix A.

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Non-Archimedean geometry of Artin fans

Ulirsch

2019

Advances in Mathematics

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The purpose of this article is to study the role of Artin fans in tropical and non-Archimedean geometry. Artin fans are logarithmic algebraic stacks that can be described completely in terms of combinatorial objects, so called Kato stacks, a stack-theoretic generalization of K. Kato's notion of a fan. Every logarithmic algebraic stack admits a tautological strict morphism φX : X → AX to an associated Artin fan. The main result of this article is that, on the level of underlying topological spaces, the natural functorial tropicalization map of X is nothing but the non-Archimedean analytic map associated to φX by applying Thuillier's generic fiber functor. Using this framework, we give a reinterpretation of the main result of Abramovich-Caporaso-Payne identifying the moduli space of tropical curves with the non-Archimedean skeleton of the corresponding algebraic moduli space. ∼ − −→ ∆ that makes the diagram X A X ∆ φ X trop ∆ ∼ µ X 4 5

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Tropical curves, graph complexes, and top weight cohomology of M_g

Cited by 15 publications

References 24 publications

Topology of moduli spaces of tropical curves with marked points

Topology of moduli spaces of tropical curves with marked points

Automorphisms of tropical Hassett spaces

Non-Archimedean geometry of Artin fans

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