  Let $Y$ be a smooth, projective curve of genus $g\geq 1$ over the complex
numbers. Let $H^0_{d,A}(Y)$ 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 $det(p_{*}O_X/O_Y)$ 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.
