2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122041Motivated by several recent results on the geometry of the moduli spaces $\bar{\Cal M}_{g,n}$ 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 ${\Cal M}_{g,n}$ 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 $1\le n\le 12$.Final version accepted for publication on the Journal of the London Mathematical Society (Section 6 of the previous version has been removed since the proof of Claim 6.3 contained a gap)Algebraic Geometry14H10, 14H45 (primary) 14E08, 14L30 (secondary)text