Abstract
Consider the Hurwitz space parameterizing covers of \({\mathbb{P}^1}\) branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmüller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmüller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.
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Arbarello E., Cornalba M., Griffiths P., Harris J.: Geometry of Algebraic Curves. Springer, New York (1985)
Bouw I., Möller M.: Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. Math. 172, 139–185 (2010)
Chen D.: Covers of elliptic curves and the moduli space of stable curves. J. Reine Angew. Math. 649, 167–205 (2010)
Chen D.: Square-tiled surfaces and rigid curves on moduli spaces. Adv. Math. 228(2), 1135–1162 (2011)
Chen, D., Möller, M.: Non-varying sums of Lyapunov exponents of Abelian differentials in low genus. arXiv:1104.3932
Chen, D., Möller, M.: Quadratic differentials in low genus: exceptional and non-varying. arXiv:1204.1707
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(3), 455–475 (1988)
Eskin, A., Kontsevich, M., Zorich, A.: Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow. arXiv:1112.5872
Eskin A., Kontsevich M., Zorich A.: Lyapunov spectrum of square-tiled cyclic covers. J. Mod. Dyn. 5(2), 319–353 (2011)
Eskin A., Okounkov A.: Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145(1), 59–103 (2001)
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)
Farkas G.: The birational type of the moduli space of even spin curves. Adv. Math. 223(2), 433–443 (2010)
Forni G., Matheus C., Zorich A.: Square-tiled cyclic covers. J. Mod. Dyn. 5(2), 285–318 (2011)
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)
Harris J., Morrison I.: Slopes of effective divisors on the moduli space of stable curves. Invent. Math. 99, 321–355 (1990)
Harris J., Morrison I.: Moduli of Curves. Springer, New York (1998)
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)
Lanneau E.: Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities. Comment Math. Helv. 79(3), 471–501 (2004)
Stankova Z.: Moduli of trigonal curves. J. Algebraic Geom. 9(4), 607–662 (2000)
Veech W.: The Teichmüller geodesic flow. Ann. Math. 124, 441–530 (1986)
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Chen, D. Covers of the projective line and the moduli space of quadratic differentials. Geom Dedicata 163, 105–125 (2013). https://doi.org/10.1007/s10711-012-9737-x
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DOI: https://doi.org/10.1007/s10711-012-9737-x