## Syzygies and the effective cone of the moduli space of curves

Gavril Farkas, University of Texas

Abstract:

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.