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