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The moduli space of quasistable spin curves

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Abstract

We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.

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References

  1. Abramovich, D., Jarvis, T.J.: Moduli of twisted spin curves. Proc. Am. Math. Soc. 131(3), 685–699 (2002)

    Article  MathSciNet  Google Scholar 

  2. Abreu, A., Pacini, M.: The universal tropical Jacobian and the skeleton of the Esteves’ universal Jacobian. Proc. London Math. Soc. 120(3), 328–369 (2020)

    Article  MathSciNet  Google Scholar 

  3. Abreu, A., Pacini, M.: The resolution of the universal Abel map via tropical geometry and applications. Adv. Math. 378 (2021)

  4. Abreu, A., Andria, S., Pacini, M., Taboada, D.: A universal tropical Jacobian over \(M_g^{trop}\). arXiv:1912.08675

  5. Abramovich, D., Caporaso, L., Payne, S.: The tropicalization of the moduli space of curves. Annales Scientifiques del’ENS. 48(4)(4), 765–809 (2015)

    MathSciNet  Google Scholar 

  6. Brannetti, S., Melo, M., Viviani, F.: On the tropical Torelli map. Adv. Math. 226(3), 2546–2586 (2011)

    Article  MathSciNet  Google Scholar 

  7. Caporaso, L.: A compactification of the universal Picard variety over the moduli space of stable curves. J. Am. Math. Soc. 7(3), 589–660 (1994)

    Article  MathSciNet  Google Scholar 

  8. Caporaso, L.: Compactifed Jacobians of Néron type Rend. Lincei Mat. Appl. 23, 213–227 (2012)

    MathSciNet  Google Scholar 

  9. Caporaso, L., Casagrande, C., Cornalba, M.: Moduli of roots of line bundles on curves. Trans. Am. Math. Soc. 359, 3733–3768 (2007)

    Article  MathSciNet  Google Scholar 

  10. Caporaso, L., Christ, K.: Combinatorics of compactified universal Jacobians. Adv. Math. 346, 1091–1136 (2019)

    Article  MathSciNet  Google Scholar 

  11. Caporaso, L., Melo, M., Pacini, M.: The tropicalization of the moduli space of spin curves. Selecta Mathematica 26 (2020)

  12. Caporaso, L., Melo, M., Pacini, M.: Tropical theta characteristics. ArXiv:2010.00894

  13. Chan, M., Galatius, S., Payne, S.: Tropical curves, graph complexes, and top weight cohomology of \(M_g\). J. Amer. Math. Soc. 34, 565–594 (2021)

    Article  MathSciNet  Google Scholar 

  14. Chan, M., Faber, C., Galatius, S., Payne, S.: The \(S_n\)-equivariant top weight Euler characteristic of \(M_{g,n}\). arXiv:1904.06367

  15. Chiodo, A.: Stable twisted curves and their r-spin structures. Ann. Inst. Fourier 58(5), 1635–1689 (2008)

    Article  MathSciNet  Google Scholar 

  16. Christ, K., Payne, S., Shen, J.: Compactified Jacobians as Mumford models. Preprint (2019), arXiv:1912.03653

  17. Cornalba, M.: Moduli of curves and theta-characteristics. In: Lectures on Riemann Surfaces: Proceedings of the College on Riemann Surfaces, I.C.T.P., Trieste, 1987, World Scientific, 560–589, 1989

  18. Esteves, E.: Compactifying the relative Jacobian over families of reduced curves. Trans. Amer. Math. Soc. 353, 3345–3095 (2001)

    Article  MathSciNet  Google Scholar 

  19. Esteves, E., Medeiros, N.: Limit canonical systems on curves with two components. Invent. Math. 149(2), 267–338 (2002)

    Article  MathSciNet  Google Scholar 

  20. Esteves, E., Pacini, M.: Semistable modifications of families of curves and compactified Jacobians. Ark. Mat. 54(1), 55–83 (2016)

    Article  MathSciNet  Google Scholar 

  21. Holmes, D., Molcho, S., Pandharipande, R., Pixton, A., Schmitt, J.: Logarithmic double ramification cycles. arXiv:2207.06778

  22. Jarvis, T.J.: Torsion-free sheaves and moduli of generalized spin curves. Compos. Math. 110, 291–333 (1998)

    Article  MathSciNet  Google Scholar 

  23. Jarvis, T.J.: Geometry of the moduli of higher spin curves. Int. J. Math. 11(5), 637–663 (2000)

    Article  MathSciNet  Google Scholar 

  24. Kass, J., Pagani, N.: The stability space of compactified universal Jacobians. Trans. Am. Math. Soc. 372, 4851–4887 (2019)

    Article  MathSciNet  Google Scholar 

  25. Melo, M.: Compactifications of the universal Jacobian over curves with marked points. arXiv:1509.06177

  26. Zharkov, I.: Tropical theta characteristics. Mirror symmetry and tropical geometry. Contemp. Math., 527, A.M.S., Providence, RI (2010), pp. 165–168

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Acknowledgements

This is a part of the Ph.D. thesis of the third author supervised by the first and second author. We thank Ethan Cotterill and Margarida Melo for some discussions and for the positive comments on a preliminary version of the paper. We also thank the anonymous referee for helpful suggestions.

Funding

Marco Pacini was supported by CNPq-PQ, 301671/2019-2. Danny Taboada was supported by Capes.

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Correspondence to Marco Pacini.

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Abreu, A., Pacini, M. & Taboada, D. The moduli space of quasistable spin curves. Collect. Math. 75, 27–80 (2024). https://doi.org/10.1007/s13348-022-00377-2

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