# Slope conjecture

The slope conjecture is about the possible homology classes of hypersurfaces in the moduli space of curves. Given an effective line bundle on , we can find non-negative for which

where is the Hodge class and is the class Poincare-dual to the boundary divisor . The slope of the divisor is . The slope conjecture states that

For this would imply that the Kodaira dimension of is . 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 . It is known that .

Jeffrey Herschel Giansiracusa 2005-06-27