Slope conjecture

The slope conjecture is about the possible homology classes of hypersurfaces in the moduli space of curves. Given an effective line bundle $L$ on $\overline{\mathcal{M}}_{g}$, we can find non-negative $a,b_{i}$ for which

$$c_{1}(L) = a\lambda - b_{0}\delta_{0} \dots -b_{[g/2]}\delta_{[g/2]} \in H^{2}(\overline{\mathcal{M}}_g,\mathbb{Q})$$

where $\lambda$ is the Hodge class $c_{1}(\mathbb{E})$ and $\delta_{i}$ is the class Poincare-dual to the boundary divisor $\Delta_{i}$. The slope of the divisor $L$ is $s(L) = \frac{a}{\mathrm{min}b_{i}}$. The slope conjecture states that

$$s_{g} = \mathrm{inf}_{L, a\neq 0}s(L) \geq 6 + \frac{12}{g+1}.$$

For $g \leq 22$ this would imply that the Kodaira dimension of $\overline{\mathcal{M}}_{g}$ is $-\infty$. As it happens, Farkas and Popa have recently constructed several counterexamples to the slope conjecture. However, one can still ask for other weaker lower bounds on $s_{g}$. It is known that $s_{g} \geq 4$.