On the rationality of moduli spaces of pointed curves

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