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Covers of the projective line and the moduli space of quadratic differentials
Consider the 1-dimensional Hurwitz space parameterizing covers of P^1
branched at four points. We study its intersection with divisor classes on the
moduli space of curves. As an application, we calculate the slope of the
Teichmuller curve parameterizing square-tiled cyclic covers and recover the sum
of its Lyapunov exponents obtained by Forni, Matheus and Zorich. Motivated by
the work of Eskin, Kontsevich and Zorich, we exhibit a relation among the slope
of Hurwitz spaces, the sum of Lyapunov exponents and the Siegel-Veech constant
for the moduli space of quadratic differentials
Square-tiled surfaces and rigid curves on moduli spaces
This work is motivated by two central questions in the birational geometry of
moduli spaces of curves -- Fulton's conjecture and the effective cone of . We study the algebro-geometric aspect of Teichmuller curves
parameterizing square-tiled surfaces with two applications:
(a) there exist infinitely many rigid curves on the moduli space of
hyperelliptic curves, they span the same extremal ray of the cone of moving
curves and their union is Zariski dense, hence they yield infinitely many rigid
curves with the same properties on the moduli space of stable n-pointed
rational curves for even n;
(b) the limit of slopes of Teichmuller curves and the sum of Lyapunov
exponents of the Hodge bundle determine each other, by which we can have a
better understanding for the cone of effective divisors on the moduli space of
curves.Comment: Section 2 is motivated by conversations with Valery Alexeev. Section
3 and Appendix A are written with the help of Alex Eskin. Appendix B is due
to Anton Zorich. The results were first announced at KIAS workshop on Moduli
and Birational Geometry, Seoul, Dec 200
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