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.
Communicated by: R. Cavalieri
Funding
This project was completed as part of the 2017 Summer Undergraduate Mathematics Research at Yale(S.U.M.R.Y.) program. D.R. was partially supported by NSF grant number DMS-1128155 (Institute for Advanced Study).
Acknowledgements
We are grateful to all the participants for helping to create a stimulating mathematical environment. The research presented here benefited from conversations with Kenny Ascher, Dori Bejleri, Melody Chan, Netanel Friedenberg, Dave Jensen, Sam Payne, and Jonathan Wise, and originates from discussions with Renzo Cavalieri, Simon Hampe, and Hannah Markwig.
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