Syzygies of curves and the effective cone of $\overline{\mathcal{M}}_g$
Gavril Farkas
Source: Duke Math. J. Volume 135, Number 1 (2006), 53-98.
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
Primary Subjects: 14H10
Secondary Subjects: 13D02
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Alternatively, the document is available for a cost of $25. Select the "buy article" button below to purchase this document from a secured VeriSign, Inc. site.
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.dmj/1159281064
Digital Object Identifier: doi:10.1215/S0012-7094-06-13512-3
Mathematical Reviews number (MathSciNet):
MR2259923
References
E. Arbarello and M. Cornalba, Footnotes to a paper of Beniamino Segre: ``On the modules of polygonal curves and on a complement to the Riemann existence theorem,'' Math. Ann. 256 (1981), 341--362.
Mathematical Reviews (MathSciNet):
MR0626954
Digital Object Identifier: doi:10.1007/BF01679702
—, Calculating cohomology groups of moduli spaces of curves via algebraic geometry, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97--127.
Mathematical Reviews (MathSciNet):
MR1733327
Digital Object Identifier: doi:10.1007/BF02701767
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. 1, Grundlehren Math. Wiss. 267, Springer, New York, 1998.
Mathematical Reviews (MathSciNet):
MR0770932
D. Eisenbud and J. Harris, Limit linear series: Basic theory, Invent. Math. 85 (1986), 337--371.
Mathematical Reviews (MathSciNet):
MR0846932
Digital Object Identifier: doi:10.1007/BF01389094
—, The Kodaira dimension of the moduli space of curves of genus $\geq 23,$ Invent. Math. 90 (1987), 359--387.
Mathematical Reviews (MathSciNet):
MR0910206
Digital Object Identifier: doi:10.1007/BF01388710
—, Irreducibility of some families of linear series with Brill-Noether number, I, Ann. Sci. École Norm. Sup. (4) 22 (1989), 33--53.
Mathematical Reviews (MathSciNet):
MR0985853
G. Farkas, The geometry of the moduli space of curves of genus $23$, Math. Ann. 318 (2000), 43--65.
Mathematical Reviews (MathSciNet):
MR1785575
Digital Object Identifier: doi:10.1007/s002080000108
G. Farkas and M. Popa, Effective divisors on $\mm_g$, curves on $K3$ surfaces, and the slope conjecture, J. Algebraic Geom. 14 (2005), 241--267.
Mathematical Reviews (MathSciNet):
MR2123229
W. Fulton, Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin, 1998.
Mathematical Reviews (MathSciNet):
MR1644323
M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compositio Math. 67 (1988), 301--314.
Mathematical Reviews (MathSciNet):
MR0959214
J. Harris and I. Morrison, Slopes of effective divisors on the moduli space of stable curves, Invent. Math. 99 (1990), 321--355.
Mathematical Reviews (MathSciNet):
MR1031904
Digital Object Identifier: doi:10.1007/BF01234422
J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), 23--88.
Mathematical Reviews (MathSciNet):
MR0664324
Digital Object Identifier: doi:10.1007/BF01393371
R. Lazarsfeld, ``A sampling of vector bundle techniques in the study of linear series'' in Lectures on Riemann Surfaces (Trieste, Italy, 1987), World Sci., Teaneck, N.J. 1989, 500--559.
Mathematical Reviews (MathSciNet):
MR1082360
A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), 105--138.
Mathematical Reviews (MathSciNet):
MR1953519
Digital Object Identifier: doi:10.1353/ajm.2003.0005
C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces, Progr. Math. 3, Birkhäuser, Boston, 1980.
Mathematical Reviews (MathSciNet):
MR0561910
F.-O. Schreyer and F. Tonoli, ``Needles in a haystack: Special varieties via small fields'' in Computations in Algebraic Geometry with Macaulay 2, Algorithms Comput. Math. 8, Springer, Berlin, 2002, 251--279.
Mathematical Reviews (MathSciNet):
MR1949554
C. Voisin, Sur l'application de Wahl des courbes satisfaisant la condition de Brill-Noether-Petri, Acta Math. 168 (1992), 249--272.
Mathematical Reviews (MathSciNet):
MR1161267
Digital Object Identifier: doi:10.1007/BF02392980
—, Green's generic syzygy conjecture for curves of even genus lying on a $K3$ surface, J. Eur. Math. Soc. (JEMS) 4 (2002), 363--404.
Mathematical Reviews (MathSciNet):
MR1941089
Digital Object Identifier: doi:10.1007/s100970200042
Duke Mathematical Journal
