Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects 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 extrernal ray of the cone of moving curves. Their union is a Zariski dense subset. 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 for the Teichmuller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves. (C) 2011 Elsevier Inc. All rights reserved.