On the Kodaira dimension of Hurwitz spaces

Abstract

The Hurwitz space H k g is the parameter space of covers [f :where C is a smooth algebraic curve of genus g and f is a degree k map simply branched over b = 2g + 2k − 2 distinct points p 1 , . . . , p b ∈ P 1 . Note that we choose an ordering of the branch points of f . The origins of the interest in Hurwitz spaces go back to Riemann's Existence Theorem and they have been used by Clebsch [Cl] and Hurwitz [Hu], as well as much later in [HM] to derive important information on the moduli space M g of curves of genus g. We denote by H g,k the moduli space of admissible covers constructed by Harris and Mumford [HM], whose study has been further refined in [ACV] via twisted stable maps. It comes equipped with two maps