Components of simple and non--simple type of Hurwitz schemes

Abstract

Let $\mathcal{H}_{g \to b,d; \mathbf{e}}$, with $\mathbf{e}=(e_1,\ldots, e_n)$, be the Hurwitz space, parametrizing all morphisms $π: C\to B$ of degree $d$, with $n$ points $x_1,\ldots, x_n\in C$ of ramification order $e_1,\ldots, e_n$ respectively, and where $C$ and $B$ are smooth, irreducible, projective curves of genera $g$ and $b$ respectively. In this paper we study the question of when there exist components of $\mathcal{H}_{g \to b,d; \mathbf{e}}$ whose members $π: C \to B$ all factor through an intermediate curve, in which case we say that these components are \emph{of non--simple type}. We give necessary and sufficient conditions for the existence of components of non--simple type. Then we prove that for $b\geq 2$ there are always components of simple type, and for $b\in \{0,1\}$ there are such components under suitable sufficient conditions. However there are easy examples for $b\in \{0,1\}$ in which there are never components of simple type.