Syzygies and the effective cone of the moduli space of curves
Gavril Farkas, University of Texas

One of the fundamental invariants of the moduli space of stable curves is its cone of effective divisors which loosely speaking determines all the rational maps from M_g to other varieties. We present a systematic way of constructing effective divisors on M_g having exceptionally small slope. In particular these divisors provide infinitely many counterexamples to the Harris-Morrison Slope Conjecture. We also introduce a new geometric stratification of M_g somewhat analogous to the classical stratification given by gonality, but where the role of hyperreliptic curves is played by curves that lie on K3 surfaces. As an application we prove that several moduli spaces of (pointed) curves are of general type.