Algebraic families of Riemann surfaces and topological invariants
Matthew Simpson (Rice)

Abstract: The Mumford-Knudsen moduli space of pointed curves parameterizes algebraic families of Riemann surfaces of a given genus along with collections of distinguished sections. Much of the information about these moduli spaces are encoded in their boundaries, which correspond to certain degenerations of smooth surfaces.

In this talk I study the cone of curves which live in the second homology of the genus zero Mumford-Knudsen moduli spaces. Studying this cone is equivalent to studying how one dimensional families of pointed spheres vary. For example, the structure of the cone of curves provides certain bounds for the type and number of degenerate fibers which may occur in such a family. In addition, extremal rays of this cone are related to extremal contractions of our moduli space via the minimal model program.

I will survey some of the methods used to compute these cones and provide, as a consequence of a conjectural description by Fulton of the cone of curves, a concrete description of certain extremal contractions of our moduli space corresponding to weighting our distinguished sections.