  We describe a systematic way of constructing effective divisors on the moduli
space of stable curves of genus g having exceptionally small slope. We prove
that any divisor on \bar{M}_g consisting of curves failing a certain
Green-Lazarsfeld syzygy type condition, provides a counterexample to the
Harris-Morrison Slope Conjecture. These divisors generalize our original
isolated counterexample to the Slope Conjecture which was the divisor on M_{10}
of curves lying on K3 surfaces. We also introduce a new stratification of M_g,
somewhat similar to the classical stratification given by gonality, but where
the analogue of hyperelliptic curves are sections of K3 surfaces. Finally, we
prove that various moduli spaces M_{g,n} with g<23 are of general type.
