Admissible covers were developed by Harris and Morrison. The space of
admissible covers is a modular compactification of the Hurwitz scheme
obtained by allowing the target
to degenerate when
branch points approach each other. The map
given by forgetting the map to
extends to a map
.
A different compactification of the Hurwitz scheme was given by
Abramovich and Vistoli. Their compactification has two advantages:
(i) it is smooth, and (ii) it is a moduli space. The disadvantage is
that is is a Deligne-Mumford stack ratehr than a scheme.
Jeffrey Herschel Giansiracusa
2005-06-27