On the rationality of moduli spaces of pointed hyperelliptic curves

Let ${\Cal M}_{g,n}$ be the (coarse) moduli space of smooth, integral, projective curves of genus $g\ge1$ with $n$ marked points defined over the complex field $\C$. We denote by ${\Cal H}_{g,n}\subseteq{\Cal M}_{g,n}$ the locus of points corresponding to curves carrying a $g^1_2$. It is known that ${\Cal H}_{g,n}$ is rational for $g=1$ and $n\le 10$, for $g=2$ and $n\le12$ and for each $g\ge3$ and $n=0$. We prove here that the same is true for each $g\ge3$ and $1\le n\le2g+8