SYZYGIES OF CURVES AND THE EFFECTIVE CONE OF Mg

The aim of this paper is to describe a systematic way of constructing effective divisors on Mg having exceptionally small slope. In particular, these divisors provide a string of counterexamples to the Harris-Morrison Slope Conjecture (cf. [HMo]). In a previous paper [FP], we showed that the divisor K10 on M10 consisting of sections of K3 surfaces contradicts the Slope Conjecture on M10. Since the moduli spaces Mg are known to behave erratically for small g and since the condition that a curve of genus g lie on a K3 surface is divisorial only for g = 10, the question remained whether K10 is an isolated example or the first in a series of counterexamples. Here we prove that any effective divisor on Mg consisting of curves satisfying a Green-Lazarsfeld syzygy type condition for a linear system residual to a pencil of minimal degree, violates the Slope Conjecture. A consequence of the existence of these effective divisors is that various moduli spacesMg,n with g ≤ 22, are proved to be of general type.