The rationality of the Weierstrass space of type (4,g)

Let $\M_g$ be the moduli space of smooth, integral curves of genus $g$ over the complex field $\Bbb C$. We denote by ${W}_{n,g}$ the locus inside $\M_g$ of $n$--gonal curves $C$ with exactly one total ramification point, the other ramification points being simple. $\overline{W}_{n,g}$ is irreducible of dimension $2g+n-3$. Moreover for $2\le n\le 5$ it is also unirational. It is then a natural question to ask whether $\overline{W}_{n,g}$ is also rational for this values of $n$. The locus $\overline{W}_{2,g}$ is the hyperelliptic locus and F\. Bogomolov and P\. Katsylo proved its rationality for any $g\ge2$. More recently we proved that $\overline{W}_{3,g}$ is rational too when $g\ge4$. In the present paper we prove that $\overline{W}_{4,g}$ is also rational when $g\ge6$.