On the rationality of moduli spaces of pointed curves

Motivated by several recent results on the geometry of the moduli spaces $\overline{\Cal M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points, we determine here their birational structure for small values of $g$ and $n$ by exploiting suitable plane models of the general curve. More precisely, we show that ${\Cal M}_{g,n}$ is rational for $g=2$ and $1\le n \le 12$, $g=3$ and $1\le n \le 14$, $g=4$ and $1 \le n \le 15$, $g=5$ and $1\le n\le 12$.