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Covers of the projective line and the moduli space of quadratic differentials. (English) Zbl 1271.14036
The 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.

##### 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
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##### 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
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