On the unirationality of moduli spaces of pointed curves

We show that ${\mathcal{M}}_{g,n}$, the moduli space of smooth curves of genus $g$ together with $n$ marked points, is unirational for $g=12$ and $2 \leq n\leq 4$andfor$g=13$and$1 \leq n \leq 3$, by constructing suitable dominant families of projective curves in ${\mathbb{P}}^1\times {\mathbb{P}}^2$ and ${\mathbb{P}}^3$ respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus $g$ together with $n$ unordered points, establishing their unirationality for $g=11,n=7$ and $g=12, n =5,6$.