Every curve is a Hurwitz space (Q923638)

The main result of this paper is that any branched covering map \(\pi\) : \(C\to {\mathbb{P}}^ 1\) ramified only over 0, 1 and \(\infty\), arises from a Hurwitz space. That is, C is the closure of an irreducible component of a Hurwitz scheme H parametrizing a family of branched covers of \({\mathbb{P}}^ 1\) branched only over 0, 1, \(\infty\), and \(\lambda\), where \(\lambda\) is allowed to vary. As applications it is shown that a nonsingular complete curve is a Hurwitz curve iff it is definable over \({\bar {\mathbb{Q}}}\), and that there exist branched covers of \({\mathbb{P}}^ 1\) branched at 4 points for which the Hurwitz number (the number of components of H) is arbitrarily large.