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Let $X$, $X^{\prime }$, $Y$ be smooth projective complex curves with $Y$ curve of genus $\ge 1$. Let $d$ be an integer $\ge 3$, let $\underset{¯}{e}=(e_1,\mathrm{\dots },e_r)$ be a partition of $d$ and let $|e|={\sum }_{i=1}^r(e_i-1)$. Let $X\stackrel{𝜋}{\to }{X}^{\prime }\stackrel{𝑓}{\to }Y$ 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 $\underset{¯}{e}$. Moreover the branch locus of the covering $\pi$ is contained in $f^{-1}(c)$. In this paper we prove the irreducibility of the Hurwitz...

Let Y be a smooth, connected, projective complex curve of genus > = 0. Biggers and Fried proved the irreducibility of the Hurwitz spaces which parametrize coverings of $P^1$whose monodromy group is a Weyl of type $W(B_d)$. Here we prove the irreducibility of Hurwitz spaces that parametrize coverings of Y with monodromy group a Weyl group of type $W(B_d)$.