## Duke Mathematical Journal

### Syzygies of curves and the effective cone of $\overline{\mathcal{M}}_g$

Gavril Farkas

#### 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 $\overline{\mathcal{M}}_g$ 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

#### Article information

Source
Duke Math. J. Volume 135, Number 1 (2006), 53-98.

Dates
First available in Project Euclid: 26 September 2006

http://projecteuclid.org/euclid.dmj/1159281064

Digital Object Identifier
doi:10.1215/S0012-7094-06-13512-3

Mathematical Reviews number (MathSciNet)
MR2259923

Zentralblatt MATH identifier
1107.14019

Subjects
Primary: 14H10: Families, moduli (algebraic)
Secondary: 13D02: Syzygies, resolutions, complexes

#### Citation

Farkas, Gavril. Syzygies of curves and the effective cone of M ̲ g . Duke Math. J. 135 (2006), no. 1, 53--98. doi:10.1215/S0012-7094-06-13512-3. http://projecteuclid.org/euclid.dmj/1159281064.

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