The moduli space of curves is an algebraic variety (more accurately a scheme or a stack) whose points parametrize isomorphism classes of algebraic curves of genus g. In studying the behavior of families of curves, it is useful to consider each curve as a point in the moduli space, or in the natural projective closure (called the Deligne-Mumford compactification) whose points correspond to curves with at most nodal singularities. The components of the boundary are the images of maps from moduli spaces of stable n-pointed curves. In fact, it turns out that the birational geometry of these various moduli spaces of curves frequently reveals interesting properties of families of curves.
One of the most fruitful ways to study the birational geometry of projective varieties is to investigate the divisors and curves on them. In particular, it is a fundamental problem to describe the effective and nef cones of divisors, and hence the effective cone of curves of these spaces. The research program to be pursued is a many faceted attack on this problem. The goal is to provide further clarification of the nature of these cones, the relationships between them as well as to provide a collection of rich examples which would deepen understanding of the birational geometry of the moduli spaces and also broaden general knowledge about cones of curves and divisors.
