The ample-cone conjecture for moduli spaces of stable curves

Let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable $n$-pointed curves of genus $g$, let $NE_1(\overline{\mathcal{M}}_{g,n})$ denote its Mori cone of curves, and let an $F$-curve be one of the standard boundary curves obtained from a four-pointed rational spine. A divisor $D$ on $\overline{\mathcal{M}}_{g,n}$ is tested against curves by its intersection number $D\cdot C$. Gibney–Keel–Morrison conjecture. The Mori cone $NE_1(\overline{\mathcal{M}}_{g,n})$ is generated by $F$-curves. A divisor $D$ on $\overline{\mathcal{M}}_{g,n}$ is ample if and only if $D\cdot C>0$ for every $F$-curve $C$ in $\overline{\mathcal{M}}_{g,n}$. The conjecture predicts that the extremal rays of the moduli space have modular meaning and would give an explicit description of its ample cone. It has been checked on $\overline{\mathcal{M}}_g$ for all $g\leq 24$ and reduced in fixed genus to a combinatorial question.