Topology of tropical moduli of weighted stable curves
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10.1515_advgeom-2019-0034.pdf
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Published version
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814.21 KB
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Author(s)
Cerbu, Alois
Marcus, Steffen
Peilen, Luke
Ranganathan, Dhruv
Salmon, Andrew
Date Issued
January 25, 2020
Journal
Advances in Geometry
Publisher
Walter de Gruyter GmbH
Citation
Cerbu, Alois, Marcus, Steffen, Peilen, Luke, Ranganathan, Dhruv and Salmon, Andrew. "Topology of tropical moduli of weighted stable curves" Advances in Geometry, vol. 20, no. 4, 2020, pp. 445-462.
Version
Final published version
Abstract
The moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1515/advgeom-2019-0034
