Covers of the projective line and the moduli space of quadratic differentials.

*(English)*Zbl 1271.14036The paper considers the \(1\)-dimensional Hurwitz spaces \(\overline{\mathcal{H}}_d({\mathbf c})\) of coverings \(\pi: C \to \mathbb{P}^1\) with prescribed branching behaviour at \(4\) points and no branching otherwise. They are finite coverings of \(\overline{M}_{0,4}\) and admit natural maps \(h:\overline{\mathcal{H}}_d({\mathbf c}) \to \overline{\mathcal{M}}_g\) that associate to \(\pi\) the source curve \(C\). The author proves an expression for the slope of the image under \(h\) of the irreducible components \(\overline{Z}_\mathcal{O}\) of \(\overline{\mathcal{H}}_d({\mathbf c})\), involving only combinatorial data. Examples are given, and the case of cyclic covers, where the monodromy datum is of the form \((\gamma^{a_1}, \gamma^{a_2}, \gamma^{a_3}, \gamma^{a_4})\) with \(\gamma \in S_d\) a cycle of length \(d\), is then worked out explicitly. The curves \(\overline{Z}_\mathcal{O}\) are Teichmüller curves, and the author derives a formula for the sum of their positive Lyapunov exponents (which was obtained previously in [G. Forni, C. Matheus and A. Zorich, J. Mod. Dyn. 5, No. 2, 285–318 (2011; Zbl 1276.37021)]).

Attention is then given to a special Hurwitz space \(\overline{\mathcal{H}}_d({\mathbf c})\), where the ramification profile \({\mathbf c}\) is chosen in such a way that the source curve admits a quadratic differential having only zeroes of odd order. These covers exist for any even degree, and an expression for its asymptotic slope as \(d \to \infty\) is given which depends on the Siegel-Veech constant of the corresponding stratum in the moduli space of quadratic differentials. Assuming that (as suggested by numerical evidence) these constants tend to \(1\) if the genus \(g\) tends to infinity, this gives an asymptotic lower bound of \(432/7g\) for the slope of the effective cone of \(\overline{\mathcal{M}}_g\) (cf. [D. Chen, G. Farkas and I. Morrison, “Effective divisors on moduli spaces of curves and abelian varieties”, arXiv:1205.6138]). As an application, this would give new asymptotic relations on the locus of Jacobians in the moduli space of abelian varieties.

Attention is then given to a special Hurwitz space \(\overline{\mathcal{H}}_d({\mathbf c})\), where the ramification profile \({\mathbf c}\) is chosen in such a way that the source curve admits a quadratic differential having only zeroes of odd order. These covers exist for any even degree, and an expression for its asymptotic slope as \(d \to \infty\) is given which depends on the Siegel-Veech constant of the corresponding stratum in the moduli space of quadratic differentials. Assuming that (as suggested by numerical evidence) these constants tend to \(1\) if the genus \(g\) tends to infinity, this gives an asymptotic lower bound of \(432/7g\) for the slope of the effective cone of \(\overline{\mathcal{M}}_g\) (cf. [D. Chen, G. Farkas and I. Morrison, “Effective divisors on moduli spaces of curves and abelian varieties”, arXiv:1205.6138]). As an application, this would give new asymptotic relations on the locus of Jacobians in the moduli space of abelian varieties.

Reviewer: Fabian Müller (Berlin)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14H30 | Coverings of curves, fundamental group |

30F30 | Differentials on Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

37F99 | Dynamical systems over complex numbers |

##### Keywords:

branched cover; moduli space of curves; effective slope; quadratic differential; Lyapunov exponent##### References:

[1] | Arbarello E., Cornalba M., Griffiths P., Harris J.: Geometry of Algebraic Curves. Springer, New York (1985) · Zbl 0559.14017 |

[2] | Bouw, I.; Möller, M., Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. Math., 172, 139-185, (2010) · Zbl 1203.37049 |

[3] | Chen, D., Covers of elliptic curves and the moduli space of stable curves, J. Reine Angew. Math., 649, 167-205, (2010) · Zbl 1208.14024 |

[4] | Chen, D., Square-tiled surfaces and rigid curves on moduli spaces, Adv. Math., 228, 1135-1162, (2011) · Zbl 1227.14030 |

[5] | Chen, D., Möller, M.: Non-varying sums of Lyapunov exponents of Abelian differentials in low genus. arXiv:1104.3932 |

[6] | Chen, D., Möller, M.: Quadratic differentials in low genus: exceptional and non-varying. arXiv:1204.1707 · Zbl 0674.14006 |

[7] | Cornalba, M.; Harris, J., Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. École Norm. Sup. (4), 21, 455-475, (1988) · Zbl 0674.14006 |

[8] | Eskin, A., Kontsevich, M., Zorich, A.: Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. arXiv:1112.5872 · Zbl 1305.32007 |

[9] | Eskin, A.; Kontsevich, M.; Zorich, A., Lyapunov spectrum of square-tiled cyclic covers, J. Mod. Dyn., 5, 319-353, (2011) · Zbl 1254.32019 |

[10] | Eskin, A.; Okounkov, A., Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, Invent. Math., 145, 59-103, (2001) · Zbl 1019.32014 |

[11] | Eskin, A., Okounkov, A.: Pillowcases and quasimodular forms. Algebraic Geometry and Number Theory, Progr. Math., vol. 253, pp. 1-25. Birkhäuser Boston, Boston, MA (2006) · Zbl 1136.14039 |

[12] | Farkas, G., The birational type of the moduli space of even spin curves, Adv. Math., 223, 433-443, (2010) · Zbl 1183.14020 |

[13] | Forni, G.; Matheus, C.; Zorich, A., Square-tiled cyclic covers, J. Mod. Dyn., 5, 285-318, (2011) · Zbl 1276.37021 |

[14] | Grushevsky, S.: Geometry of \({\mathcal{A}_g}\) and its compactifications. Algebraic geometry-Seattle 2005. Part 1, 193-234, Proc. Symp. Pure Math., 80, Part 1, Am. Math. Soc., Providence, RI (2009) · Zbl 1171.14026 |

[15] | Harris, J.; Morrison, I., Slopes of effective divisors on the moduli space of stable curves, Invent. Math., 99, 321-355, (1990) · Zbl 0705.14026 |

[16] | Harris J., Morrison I.: Moduli of Curves. Springer, New York (1998) · Zbl 0913.14005 |

[17] | Kontsevich, M.: Lyapunov Exponents and Hodge Theory, The Mathematical Beauty of Physics (Saclay, 1996) Adv. Ser. Math. Phys., vol. 24, pp. 318-332. World Sci. Publ., River Edge, NJ (1997) · Zbl 1058.37508 |

[18] | Lanneau, E., Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities, Comment Math. Helv., 79, 471-501, (2004) · Zbl 1054.32007 |

[19] | Stankova, Z., Moduli of trigonal curves, J. Algebraic Geom., 9, 607-662, (2000) · Zbl 1001.14007 |

[20] | Veech, W., The Teichmüller geodesic flow, Ann. Math., 124, 441-530, (1986) · Zbl 0658.32016 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.