Abstract
We study the topology of moduli spaces of weighted stable tropical curves Δg,w with fixed genus and unit volume. The space Δg,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space Mg,w of weighted stable curves. When the genus is positive, we show that Δg,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg,w in terms of the combinatorics of the weight vector.
Funding statement: SK was supported by an NSF Graduate Research Fellowship.
Acknowledgements
We are grateful to Melody Chan for suggesting a motivic approach to the proof of Theorem 1.2, to Sam Payne for help with questions about mixed Hodge structures and connections to geometric group theory, and to Sam Freedman for many useful conversations related to this work. We also thank the organizers of the 2020 Summer Tropical Algebraic Geometry Online SeminAUR (STAGOSAUR)/Algebraic and Tropical Online Meetings (ATOM) for great learning opportunities and for bringing us together.
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Communicated by: R. Cavalieri
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