On the birational geometry of moduli spaces of pointed curves

Abstract

We prove that the moduli space ${\Cal M}_{g,n}$ of smooth curves of genus $g$ with $n$ marked points is rational for $g=6$ and $1\le n\le 8$, and it is unirational for $g=8$ and $1\le n\le 11$, $g=10$ and $1\le n\le 3$, $g=12$ and $n=1$.