Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T15:11:30.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Green’s conjecture for curves on arbitrary K3 surfaces

Part of: Commutative algebra: Homological methods Cycles and subschemes

Published online by Cambridge University Press:  15 February 2011

Marian Aprodu  and
Gavril Farkas
Marian Aprodu
Affiliation:
Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, RO-014700 Bucharest, Romania (email: [email protected]) Şcoala Normală Superioară Bucureşti, Calea Griviţei 21, Sector 1, RO-010702 Bucharest, Romania
Gavril Farkas
Affiliation:
Humboldt-Universität zu Berlin, Institut Für Mathematik, 10099 Berlin, Germany (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Green’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

Keywords

syzygycanonical curveBrill–Noether theory

MSC classification

Secondary: 13D02: Syzygies, resolutions, complexes 14C20: Divisors, linear systems, invertible sheaves
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 3 , May 2011 , pp. 839 - 851
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[1]Aprodu, M., On the vanishing of higher syzygies of curves, Math. Z. 241 (2002), 1–15.Google Scholar
[2]Aprodu, M., Remarks on syzygies of d-gonal curves, Math. Res. Lett. 12 (2005), 387–400.CrossRefGoogle Scholar
[3]Aprodu, M. and Nagel, J., Koszul cohomology and algebraic geometry, University Lecture Series, vol. 52 (American Mathematical Society, Providence, RI, 2010).Google Scholar
[4]Aprodu, M. and Pacienza, G., The Green conjecture for exceptional curves on a K3 surface, Int. Math. Res. Not. IMRN (2008), 25.Google Scholar
[5]Aprodu, M. and Voisin, C., Green–Lazarsfeld’s conjecture for generic curves of large gonality, C. R. Math. Acad. Sci. Paris 36 (2003), 335–339.Google Scholar
[6]Ciliberto, C. and Pareschi, G., Pencils of minimal degree on curves on a K3 surface, J. Reine Angew. Math. 460 (1995), 15–36.Google Scholar
[7]Donagi, R. and Morrison, D. R., Linear systems on K3-sections, J. Differential Geom. 29 (1989), 49–64.Google Scholar
[8]Eisenbud, D., Lange, H., Martens, G. and Schreyer, F.-O., The Clifford dimension of a projective curve, Compositio Math. 72 (1989), 173–204.Google Scholar
[9]Green, M., Koszul cohomology and the geometry of projective varieties, J. Differential Geom. 19 (1984), 125–171.CrossRefGoogle Scholar
[10]Green, M. and Lazarsfeld, R., On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90.Google Scholar
[11]Green, M. and Lazarsfeld, R., Special divisors on curves on a K3 surface, Invent. Math. 89 (1987), 357–370.Google Scholar
[12]Hirschowitz, A. and Ramanan, S., New evidence for Green’s conjecture on syzygies of canonical curves, Ann. Sci. École Norm. Sup. (4) 31 (1998), 145–152.Google Scholar
[13]Knutsen, A., On two conjectures for curves on K3 surfaces, Internat. J. Math. 20 (2009), 1547–1560.Google Scholar
[14]Lazarsfeld, R., Brill–Noether–Petri without degenerations, J. Differential Geom. 23 (1986), 299–307.Google Scholar
[15]Lazarsfeld, R., A sampling of vector bundle techniques in the study of linear series, in Proceedings of the first college on Riemann surfaces, Trieste, November 9–December 18, 1987, eds Cornalba, M.et al. (World Scientific, Teaneck, NJ, 1989), 500–559.Google Scholar
[16]Martens, G., On curves on K3 surfaces, in Algebraic curves and projective geometry, Trento, 1988, Lecture Notes in Mathematics, vol. 1389 (Springer, Berlin–New York, 1989), 174–182.Google Scholar
[17]Pareschi, G., A proof of Lazarsfeld’s theorem on curves on K3 surfaces, J. Algebraic Geom. 4 (1995), 195–200.Google Scholar
[18]Saint-Donat, D., Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602–639.Google Scholar
[19]Voisin, C., Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface, J. Eur. Math. Soc. (JEMS) 4 (2002), 363–404.Google Scholar
[20]Voisin, C., Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163–1190.CrossRefGoogle Scholar