The tropicalization of the moduli space of curves II: Topology and applications

Chan, Melody; Galatius, Soren; Payne, Sam

doi:10.48550/arxiv.1604.03176

Cited by 10 publications

(19 citation statements)

References 29 publications

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“…In another direction related to the results of this paper, tropicalization of moduli spaces of curves has been studied by Abramovich-Caporaso-Payne [ACP15], and in the subsequent work of Chan-Galatius-Payne in the study of the cohomology of the moduli space of curves [CGP16,CGP18]. Tropical moduli spaces have been used in [OO18,Oda19], in connection with the constructions of Boucksom and Jonsson, to construct compactifications of moduli spaces of curves and K3 surfaces.…”

Section: Consider Now a Hybrid Curve Cmentioning

confidence: 93%

Moduli of hybrid curves and variations of canonical measures

Amini¹,

Nicolussi²

2020

Preprint

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We introduce the moduli space of hybrid curves as the hybrid compactification of the moduli space of curves thereby refining the one obtained by Deligne and Mumford. As the main theorem of this paper we then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously.On the way to achieve this, we present constructions and results which we hope could be of independent interest. In particular, we introduce variants of hybrid spaces which refine and combine both the ones considered by Berkovich, Boucksom and Jonsson, and metrized complexes of varieties studied by Baker and the first named author. Furthermore, we introduce canonical measures on hybrid curves which simultaneously generalize the Arakelov-Bergman measure on Riemann surfaces, Zhang measure on metric graphs, and Arakelov-Zhang measure on metrized curve complexes.This paper is part of our attempt to understand the precise link between the Zhang and Arakelov measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.

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Section: Consider Now a Hybrid Curve Cmentioning

confidence: 93%

Moduli of hybrid curves and variations of canonical measures

Amini¹,

Nicolussi²

2020

Preprint

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“…Theorem 1.1 can also be used to deduce a lower bound on connectivity of the spaces ∆ g,n . We do not pursue this here, but refer to [CGP16,Theorem 1.3]. for all k.…”

Section: Ind Snmentioning

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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“…The above Theorem D also contributes to the study of tropical moduli spaces, which have received considerable interest in recent years. These results have aimed at an improved conceptual understanding of information contained in tropical moduli spaces, including applications to enumerative geometry [10,11,28] and to the geometry and topology of moduli spaces [9,12,13]. The novelty of the present paper is that the tropicalization of L • Γ (X) is studied as the tropicalization of a map to a certain toroidal stack -the stack of prestable maps to the Artin fan.…”

Section: Further Discussionmentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

Ranganathan

2017

Res Math Sci

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We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov-Witten theory-the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition. BackgroundCentral to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a "synthetic" tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in R n arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in R n encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same frame- 0123456789().,-: vol

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The tropicalization of the moduli space of curves II: Topology and applications

Cited by 10 publications

References 29 publications

Moduli of hybrid curves and variations of canonical measures

Moduli of hybrid curves and variations of canonical measures

Topology of moduli spaces of tropical curves with marked points

Skeletons of stable maps II: superabundant geometries

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