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

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

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

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

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$.

Jeffrey Herschel Giansiracusa 2005-06-27