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

![$c_1(L)=a\lambda -b_0{\delta }_0\cdots -b_{[g/2]}{\delta }_{[g/2]}\in H^2({\overline{\mathcal{M}}}_g,\mathbb{Q})$

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 $s(L)=\frac{a}{\min b_i}$. The slope conjecture states that

$$s_g={\inf }_{L,a\ne 0}s(L)\ge 6+\frac{12}{g+1}.$$

For $g\le 22$ 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 $s_g\ge 4$.