The Harris–Morrison slope conjecture for the moduli space of curves

Let $g\beq 4$, let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $s(\overline{\mathcal{M}}_g)$ denote its slope, defined as the infimum of the slopes of effective divisors on $\overline{\mathcal{M}}_g$. Harris–Morrison slope conjecture.

$$s(\overline{\mathcal{M}}_g)\geq 6+\frac{12}{g+1}.$$

This conjecture gives a lower bound for the slopes of effective divisors on the moduli space and has important consequences for the birational geometry of $\overline{\mathcal{M}}_g$. It is false: counterexamples were found by Farkas and others.