Algebraic Geometry Seminar
Matthew Simpson
Rice University
Log Canonical Models of the Deligne-Mumford Moduli Space of Pointed Genus Zero Stable Curves
Abstract: The study of moduli spaces is central to the understanding of the birational
geometry of varieties, classifying both the varieties themselves, and how
they vary in reasonable families. To understand all of the consequences of the
existence of a particular moduli space, one must completely understand the
geometry of the moduli space itself. One open problem in the much-studied
Deligne-Mumford Moduli Spaces of stable genus g pointed curves is to
understand the canonical--or more generally log-canonical--models.
In this talk, we examine the log-canonical models in the genus zero
case, with respect to the standard log canonical divisors $K+cD$ where $K$ is the canonical
class, and $D$ the boundary divisor parameterizing nodal curves. We will show
that a conjectural description of the cone of curves by Fulton implies that
these log-canonical models are isomorphic to Hassett's moduli space of
weighted genus zero stable curves for various weights.
For certain values $c$, we can prove this result unconditionally. For
large $c$, the proof is essentially inductive. For small $c$, we use methods
from geometric invariant theory. We will survey these results and, if given
time, discuss the consequences of the second technique to Fulton's conjecture.
Thursday November 29, 2007 at 4:00 PM in SEO 636