On the rationality of certain Weierestrass spaces of type (5,g)

Let $\M_g$ be the moduli space of smooth, integral curves of genus $g$ over the complex field $\Bbb C$ and denote by $\overline{W}_{d,g}\subseteq\M_g$ the Weierstrass space of type $(d,g)$, i.e. the closure of the locus of points representing $d$--gonal with exactly one total ramification point, the other ramification points being simple. The locus $\overline{W}_{d,g}$ is always irreducible and it is rational for $d=2,3,4$. We prove here the rationalilty of $\overline{W}_{5,g}$ for $g\equiv 16 \pmod{20}$.