The slope $s_g$ is computed by the quotient of the lambda and nonseparating-boundary coefficients

Let $\overline{\mathcal{M}}_g$ be the moduli space of stable curves of genus $g$, and let $s_g$ be its slope. For an effective divisor $D$ on $\overline{\mathcal{M}}_g$, write

$$D\equiv a\lambda-\sum_{i=0}^{\lfloor g/2\rfloor}b_i\delta_i,$$

where $\lambda$ is the Hodge class and the $\delta_i$ are the boundary divisor classes. Its slope is denoted by $s(D)$. Slope quotient conjecture. The statement of the theorem holds in arbitrary genus: for every genus $g$, there exists $\epsilon_g>0$ such that every effective divisor $D$ on $\overline{\mathcal{M}}_g$ satisfying

$$s_g\leq s(D)\leq s_g+\epsilon_g$$

has

$$s(D)=\frac{a}{b_0},\qquad b_0\leq b_i\text{ for all }i\geq 1.$$

The theorem in the paper establishes this assertion for $g\leq 23$; the displayed statement is the authors' conjectural extension to arbitrary genus.