Tau functions, Prym-Tyurin classes and loci of degenerate differentials

Abstract

We study the rational Picard group of the projectivized moduli space \(P\overline{{\mathfrak {M}}}_{g}^{(n)}\) of holomorphic n-differentials on complex genus g stable curves. We define \(n-1\) natural classes in this Picard group that we call Prym-Tyurin classes. We express these classes as linear combinations of boundary divisors and the divisor of n-differentials with a double zero. We give two different proofs of this result, using two alternative approaches: an analytic approach that involves the Bergman tau function and its vanishing divisor and an algebro-geometric approach that involves cohomological computations on the universal curve.

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Acknowledgements

DK and PZ acknowledge the hospitality of the Max-Planck-Institut für Mathematik in Bonn where this work began in 2011, and DK also thanks the Max-Planck Institute for Gravitational Physics (Albert Einstein Institute) in Golm where this work was continued. Research of DK was supported in part by the Natural Sciences and Engineering Research Council of Canada grant RGPIN/3827-2015, by the FQRNT grant “Matrices Aléatoires, Processus Stochastiques et Systémes Intégrables” (2013PR 166790), by the Alexander von Humboldt Stiftung and by the Chebyshev Laboratory of St. Petersburg State University. The work of PZ was partially supported by the Government of Russian Federation megagrant 14.W03.31.0030. Research of Sect. 3 was supported by the Russian Science Foundation grant 16-11-10039. The authors are very grateful to Dimitri Zvonkine for his helpful advice and comments. AS is also grateful to Dawei Chen, Charles Fougeron, Martin Möller and Anton Zorich for very fruitful conversations.

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Correspondence to Adrien Sauvaget.

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Korotkin, D., Sauvaget, A. & Zograf, P. Tau functions, Prym-Tyurin classes and loci of degenerate differentials. Math. Ann. 375, 213–246 (2019). https://doi.org/10.1007/s00208-019-01836-1

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Keywords

  • Moduli space of curves
  • n-differentials
  • Cyclic covers
  • Bergman tau function
  • Integrable systems

Mathematics Subject Classification

  • 14H15
  • 14F10
  • 14H70
  • 30F30
  • 14C22