# Square-tiled surfaces and rigid curves on moduli spaces

###### Abstract.

We study the algebro-geometric aspect of Teichmüller curves parameterizing square-tiled surfaces with two applications:

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 -pointed rational curves for even ;

the limit of slopes of Teichmüller 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.

###### Contents

## 1. Introduction

Let be a partition of for . The moduli space of Abelian differentials parameterizes pairs , where is a smooth genus curve, is a holomorphic 1-form and for distinct points on . The space is a complex orbifold of dimension and the period map yields its local coordinates [K]. It may have up to three connected components [KZ], corresponding to hyperelliptic, odd or even spin structures.

Consider a degree connected cover from a genus curve to the standard torus with a unique branch point , such that . Then admits a holomorphic 1-form and . It is known [EO, Lemma 3.1] that such a pair has integer coordinates under the period map. Varying the complex structure of , we obtain a Teichmüller curve passing through , which is invariant under the natural SL action on . One can regard as the 1-dimensional Hurwitz space parameterizing degree , genus connected covers of elliptic curves with a unique branch point and the ramification class . Abuse our notation and still use to denote the compactification of this Hurwitz space in the sense of admissible covers [HM2, 3.G]. The boundary points of parameterize admissible covers of rational nodal curves. We call them cusps of . Note that may be reducible. There is a monodromy criterion [C, Theorem 1.18] to distinguish its irreducible components, which correspond to the orbits of the SL action. Let be the number of irreducible components of and label these components as , . If is contained in the hyperelliptic component of , we denote it by

Our motivation is to use to study the birational geometry of moduli spaces of stable pointed rational curves and stable genus curves. Let denote the moduli space of stable -pointed genus curves. In particular, let () denote the moduli space of stable (unordered) -pointed rational curves. There are two natural morphisms as follows: