Singular Curves on K3 Surfaces
DOI:
https://doi.org/10.5644/SJM.06.2.02Keywords:
Clifford index, Koszul cohomology, Green’s conjecture, torsionfree sheafAbstract
We investigate the Clifford index of singular curves on K3 surfaces by following the lines of [10]. As a consequence, we are able to deduce from [3] that Green's conjecture holds for all integral curves on K3 surfaces.
2000 Mathematics Subject Classification. 14H51
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References
A. B. Altman, A. Iarrobino and S. L. Kleiman, Irreducibility of the compactified Jacobian. Real and complex singularities, (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), pp. 1–12. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.
M. Aprodu, Remarks on syzygies of d-gonal curves, Math. Res. Lett., 2 (2005), 387–400.
M. Aprodu and G. Farkas, Green’s conjecture for curves on arbitrary K3 surfaces, Pre-print arXiv:0911.5310 (2009).
M. Aprodu and J. Nagel, A Lefschetz type result for Koszul cohomology, Manuscripta Math., 114 (2004), 423–430.
E. Ballico, Brill-Noether theory for rank 1 torsion free sheaves on singular projective curves, J. Korean Math. Soc., 37 (2000), 359–369.
E. Ballico, C. Fontanari and L. Tasin, Koszul cohomology and singular curves, Rend. Circ. Mat. Palermo, 59 (2010), 121–125.
W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge University press, 1993.
D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlang, 1995.
M. Green, Koszul cohomology and the geometry of projective varieties, J. Diff. Geom., 19 (1984), 125–171.
M. Green and R. Lazarsfeld, Special divisors on a K3 surface, Invent. Math., 89 (1987), 357–370.
R. Hartshorne, Stable reflexive sheaves, Math. Ann., 254 (1980), 121–176.
M. Leyenson, On the Brill-Noether theory for K3 surfaces II, arXiv:math/0602358 (2006).
J.-Y. M´erindol, Propri´et´es el´ementaires des surfaces K3, Ast´erisque, 126 (1985), 45–57.
C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface, J. Eur. Math. Soc., (JEMS), 4 (2002), 363–404.
C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math., 141 (2005), 1163–1190.
