Title: Moduli space of bicanonical curves

Abstract: In this talk, I will present the Geometric Invariant Theory
of bicanonically embedded curves. A classic result of Mumford and
Gieseker is that if a curve C is embedded by n-canonical system
and n >= 5, the Hilbert point and the Chow point of C is GIT stable
if and only if C is Deligne-Mumford stable. We investigate the stability
question for the n = 2 case, and find that a semistable curve is allowed 
to have worse singularities (cusps and tacnodes) but the behaviour of its
elliptic components is more restricted.

After describing the semitable bicanonical curves and some computational
techniques, I will explain how these new moduli spaces come into play in 
the study of birational geometry of the moduli space of stable curves 
$\bar M_g$.