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Syzygies of curves and the effective cone of \bar{M}_g

Gavril Farkas
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Abstract

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.

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Gavril Farkas. 2006-12-07. Syzygies of curves and the effective cone of \bar{M}_g. https://arxiv.org/abs/math/0503498

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