(Uni)rational parametrizations of ${\mathcal{R}}_{g,2}$, ${\mathcal{R}}_{g,4}$ and ${\mathcal{R}}_{g,6}$ in low genera

The moduli space R_{g,2n} parametrizes double covers of smooth curves of genus g ramified at 2n points. We will prove the (uni)rationality of R_{g,2}, R_{g,4} and R_{g,6} in low genera.