Computation of some moduli spaces of covers and explicit $\mathcal S_n$ and $\mathcal A_n$ regular $\QQ (T)$ extensions with totally real fibers

Abstract. We study the arithmetic of a moduli space parameterizing a family of covers of $\mathbb P^1$. Our study leads to the computation of the universal curve associated to our family. We obtain this way explicit regular realizations of $\mathcal{S}_n$ and $\mathcal{A}_n$-extensions of $\QQ(x)$ with an interval of totally real fibers.