Connected Components of Hurwitz Spaces of Coverings with One Special Fiber and Monodromy Groups Contained in a Weyl Group of Type $B_d$

Vetro, Francesca

Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 1, pp. 87-103

Résumé

Let $X$, $X'$, $Y$ be smooth projective complex curves with $Y$ curve of genus $\geq 1$. Let $d$ be an integer $\geq 3$, let $\underline{e} = (e_{1}, \ldots, e_{r})$ be a partition of $d$ and let $|e|= \sum_{i=1}^{r}(e_i - 1)$. Let $X \xrightarrow{\pi} X' \xrightarrow{f} Y$ be a sequence of coverings where $\pi$ is a degree 2 branched covering and $f$ is a degree $d$ covering, with monodromy group $S_d$, branched in $n_2 + 1$ points, one of which is special point $c$ whose local monodromy has cycle type given by $\underline{e}$. Moreover the branch locus of the covering $\pi$ is contained in $f^{-1} (c)$. In this paper we prove the irreducibility of the Hurwitz spaces that parameterize sequences of coverings as above with monodromy group a Weyl group of type $D_d$ when $n_2 +|\underline{e}| \geq 2d$. Besides we determine the connected components of the Hurwitz spaces that parameterize sequences of coverings as above but with monodromy group a Weyl group of type $B_d$.