The Kodaira classification of the moduli space of pointed curves in genus $3$

We complete the Kodaira classification of the moduli spaces ${\bar{\mathcal{M}}}_{g,n}$ of curves with marked points in genus $g=3$, by proving that ${\bar{\mathcal{M}}}_{3,n}$ is of general type for $n\ge 15$. We prove that the singularities of ${\bar{\mathcal{M}}}_{3,n}$ impose no adjunction conditions for $n\ge 1$ and that the canonical class of ${\bar{\mathcal{M}}}_{3,n}$ is big for $n\ge 15$.