Covers of the projective line and the moduli space of quadratic differentials

Abstract

Consider the Hurwitz space parameterizing covers of \({\mathbb{P}^1}\) branched at four points. We study its intersection with divisor classes on the moduli space of curves. As applications, we calculate the slope of Teichmuller curves parameterizing square-tiled cyclic covers. In addition, we come up with a relation among the slope of Teichmuller curves, the sum of Lyapunov exponents and the Siegel–Veech constant for the moduli space of quadratic differentials, which yields information for the effective cone of the moduli space of curves.