Unirationality of Hurwitz spaces of coverings of degree <= 5

Let $Y$ be a smooth, projective curve of genus $g\ge 1$ over the complex numbers. Let $H_{d,A}^0(Y)$ be the Hurwitz space which parametrizes coverings $p:X\to Y$ of degree $d$, simply branched in $n=2e$ points, with monodromy group equal to $S_d$, and $det(p_{\ast }{O}_X\mathrm{/}{O}_Y)$ isomorphic to a fixed line bundle $A^{-1}$ of degree $-e$. We prove that, when $d=3,4$ or $5$ and $n$ is sufficiently large (precise bounds are given), these Hurwitz spaces are unirational. If in addition $(e,2)=1$ (when $d=3$), $(e,6)=1$ (when $d=4$) and $(e,10)=1$ (when $d=5$), then these Hurwitz spaces are rational.