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Square-tiled surfaces and rigid curves on moduli spaces. (English) Zbl 1227.14030
This paper studies the rigidity of algebraic (Teichmüller) curves arising from square tiled surfaces in the moduli space of curves. It gives a beautiful relation between the limits of slopes for Teichmüller curves and the sums of Lyapunov exponents for the Teichmüller geodesic flow.

MSC:
14H15 Families, moduli of curves (analytic)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
30F60 Teichmüller theory for Riemann surfaces
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[1] S. Boucksom, J.-P. Demailly, M. Paun, T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., in press, arXiv:math/0405285. · Zbl 1267.32017
[2] Bouw, I.; Möller, M., Teichmüller curves, triangle groups, and Lyapunov exponents, Ann. of math., 172, 139-185, (2010) · Zbl 1203.37049
[3] Castravet, A.-M.; Tevelev, J., Exceptional loci on \(\overline{M}_{0, n}\) and hypergraph curves
[4] Chen, D., Covers of elliptic curves and the moduli space of stable curves, J. reine angew. math., 649, 167-205, (2010) · Zbl 1208.14024
[5] A. Eskin, M. Kontsevich, A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, preprint. · Zbl 1305.32007
[6] Eskin, A.; Masur, H., Pointwise asymptotic formulas on flat surfaces, Ergodic theory dynam. systems, 21, 2, 443-478, (2001) · Zbl 1096.37501
[7] Eskin, A.; Masur, H.; Zorich, A., Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. math. inst. hautes études sci., 97, 61-179, (2003) · Zbl 1037.32013
[8] Eskin, A.; Okounkov, A., Asymptotic 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
[9] Gibney, A., Numerical criteria for divisors on \(\overline{\mathcal{M}}_g\) to be ample, Compos. math., 145, 5, 1227-1248, (2009) · Zbl 1184.14042
[10] Grushevsky, S., Geometry of \(\mathcal{A}_g\) and its compactifications, (), 193-234 · Zbl 1171.14026
[11] Gutkin, E.; Judge, C., Affine mappings of translation surfaces: geometry and arithmetic, Duke math. J., 103, 2, 191-213, (2000) · Zbl 0965.30019
[12] Harris, J.; Morrison, I., Slopes of effective divisors on the moduli space of stable curves, Invent. math., 99, 321-355, (1990) · Zbl 0705.14026
[13] Harris, J.; Morrison, I., Moduli of curves, (1998), Springer-Verlag New York · Zbl 0913.14005
[14] Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. math., 173, 2, 316-352, (2003) · Zbl 1072.14014
[15] Keel, S.; McKernan, J., Contractible extremal rays on \(\overline{M}_{0, n}\) · Zbl 1322.14050
[16] Kontsevich, M., Lyapunov exponents and Hodge theory, (), 318-332 · Zbl 1058.37508
[17] Kontsevich, M.; Zorich, A., Connected components of the moduli spaces of abelian differentials with prescribed singularities, Invent. math., 153, 3, 631-678, (2003) · Zbl 1087.32010
[18] McMullen, C., Rigidity of Teichmüller curves, Math. res. lett., 16, 4, 647-649, (2009) · Zbl 1187.32010
[19] Möller, M., Variations of Hodge structures of a Teichmüller curve, J. amer. math. soc., 19, 2, 327-344, (2006) · Zbl 1090.32004
[20] I. Morrison, Mori theory of moduli spaces of stable curves, preprint.
[21] R. Pandharipande, Descendent bounds for effective divisors on the moduli space of curves, J. Algebraic Geom., in press, http://www.ams.org/journals/jag/0000-000-00/S1056-3911-2010-00554-1/home.html.
[22] Veech, W., Geometric realizations of hyperelliptic curves, (), 217-226 · Zbl 0859.30039
[23] Veech, W., Siegel measures, Ann. of math. (2), 148, 3, 895-944, (1998) · Zbl 0922.22003
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