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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 $\bar M_g$. 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.