We prove the irreducibility of the space parametrizing branched covers of a fixed Riemann surface B of degree d, with at least 2d branch points, and with monodromy group equal to Sd. The result is classical for $g(B)=0$. The result is well-known for $g(B) > 0$, but we could find no reference.
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Graber et al. (2002) studied this question.