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Spin and hyperelliptic structures of log twisted differentials

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Abstract

Using stable log maps, we introduce log twisted differentials extending the notion of abelian differentials to the Deligne–Mumford boundary of stable curves. The moduli stack of log twisted differentials provides a compactification of the strata of abelian differentials. The open strata can have up to three connected components, due to spin and hyperelliptic structures. We prove that the spin parity can be distinguished on the boundary of the log compactification. Moreover, combining the techniques of log geometry and admissible covers, we introduce log twisted hyperelliptic differentials, and prove that their moduli stack provides a toroidal compactification of the hyperelliptic loci in the open strata.

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Acknowledgements

The authors thank Dan Abramovich, Matt Bainbridge, Gavril Farkas, Quentin Gendron, Samuel Grushevsky, Jérémy Guéré, Felix Janda, Martin Möller, Rahul Pandharipande, Adrien Sauvaget and Jonathan Wise for stimulating discussions on related topics. The authors also thank the referees for a number of helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Boston College, Chestnut Hill, MA, 02467, USA

    Dawei Chen & Qile Chen

  2. School of Mathematics, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ, 08540, USA

    Dawei Chen

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  1. Dawei Chen
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  2. Qile Chen
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Correspondence to Dawei Chen.

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D. Chen is supported by the NSF CAREER Grant DMS-1350396 and a von Neumann Fellowship at IAS in Spring 2019. Q. Chen is supported by the NSF Grants DMS-1560830 and DMS-1700682.

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Chen, D., Chen, Q. Spin and hyperelliptic structures of log twisted differentials. Sel. Math. New Ser. 25, 20 (2019). https://doi.org/10.1007/s00029-019-0467-x

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