Syzygies of curves and the effective cone of M̲g

Abstract

We describe a systematic way of constructing effective divisors on the moduli space of stable curves having exceptionally small slope. We show that every codimension 1 locus in ${\stackrel{̲}{M}}_g$ consisting of curves failing to satisfy a Green-Lazarsfeld syzygy-type condition provides a counterexample to the Harris-Morrison slope conjecture. We also introduce a new geometric stratification of the moduli space of curves somewhat similar to the classical stratification given by gonality but where the analogues of hyperelliptic curves are the sections of $K3$ surfaces