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.

Abstract (VO)

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 $B_d$.

Riassunto

In questo articolo vengono studiati rivestimenti $X \xrightarrow{\pi} X' \xrightarrow{f} Y$ dove $X$, $X'$, $Y$ sono curve proiettive complesse non singolari e $f$ è un rivestimento di grado $d \geq 3$, con gruppo di monodromia $S_d$, ramificato in $n_2+1$ punti uno dei quali è un punto speciale $c$ la cui monodromia locale ha struttura ciclica data dalla partizione $\underline{e} = (e_1, \ldots, e_r )$ di $d$. Inoltre $\pi$ è un rivestimento ramificato di grado 2 con luogo discriminante contenuto in $f^{-1} (c)$. Se si suppone $n_2 + |\underline{e}| \geq 2d$ dove $|\underline{e}|= \sum_{i=1}^{r}(e_i - 1)$ questi rivestimenti hanno come gruppo di monodromia $G$ un gruppo di Weyl di tipo $D_d$ oppure $B_d$. In questo articolo viene dimostrato che quando $G = W(D_d)$ e $n_2 +|\underline{e}| \geq 2d$ gli spazi di Hurwitz che parametrizzano rivestimenti come sopra sono irriducibili, mentre quando $G=W(B_d)$ non lo sono e, in quest'ultimo caso, ne vengono determinate le componenti connesse. In questo modo viene completato lo studio degli spazi di Hurwitz che parametrizzano rivestimenti con una fibra speciale e con gruppo di monodromia un gruppo di Weyl di tipo $W(B_d)$ iniziato in [22].

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