Towards a canonical model for the moduli space of curves
(joint with D. Hyeon and M. Simpson)
Consider the moduli space of pointed stable curves as a
log-variety, with boundary \delta corresponding
to the nodal curves. We seek to describe its log
canonical model with respect to K + A\delta.
When A=1, we recover the moduli space of stable curves;
for A=0, this would be the canonical model of the moduli
space, which is expected to exist for g>>0 after work of
Eisenbud-Harris-Mumford and Farkas. For some intermediate
values of A, the log canonical model can be constructed
with Geometric Invariant Theory and other techniques.
Examples of spaces that arise include D. Schubert's
moduli spaces of pseudostable curves (with nodes and
cusps), parameter spaces for bicanonical curves (allowing
tacnodes as well), and moduli spaces of weighted pointed
stable curves of genus zero.