Comptes Rendus
no. 4
Algebraic Geometry
Green–Lazarsfeld's conjecture for generic curves of large gonality
[La conjecture de Green–Lazarsfeld pour les courbes génériques de gonalité élevée]

Présenté par : Jean-Pierre Demailly

Marian Aprodu1, 2 ; Claire Voisin3
1 Université de Grenoble 1, laboratoire de mathématiques, institut Fourier, BP 74, 38402 Saint Martin d'Hères cedex, France
2 Romanian Academy, Institute of Mathematics “Simion Stoilow”, PO Box 1-764, 70700, Bucharest, Romania
3 Université Paris 7 Denis Diderot, CNRS UMR 7586, institut de mathématiques, 2, place Jussieu, 75251 Paris cedex 05, France
Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 335-339.

Nous utilisons la conjecture de Green sur les syzygies canoniques des courbes génériques pour démontrer la conjecture de la gonalité de Green–Lazarsfeld pour les courbes génériques de genre g et gonalité d, avec g/3<d<[g/2]+2.

We use Green's canonical syzygy conjecture for generic curves to prove that the Green–Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if g/3<d<[g/2]+2.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00062-1
Marian Aprodu 1, 2 ; Claire Voisin 3

1 Université de Grenoble 1, laboratoire de mathématiques, institut Fourier, BP 74, 38402 Saint Martin d'Hères cedex, France
2 Romanian Academy, Institute of Mathematics “Simion Stoilow”, PO Box 1-764, 70700, Bucharest, Romania
3 Université Paris 7 Denis Diderot, CNRS UMR 7586, institut de mathématiques, 2, place Jussieu, 75251 Paris cedex 05, France 
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Marian Aprodu; Claire Voisin. Green–Lazarsfeld's conjecture for generic curves of large gonality. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 335-339. doi : 10.1016/S1631-073X(03)00062-1. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00062-1/

[1] M. Aprodu On the vanishing of higher syzygies of curves, Math. Z., Volume 241 (2002), pp. 1-15

[2] M. Boratyńsky; S. Greco Hilbert functions and Betti numbers in a flat family, Ann. Mat. Pura Appl. (4), Volume 142 (1985), pp. 277-292

[3] S. Ehbauer Syzygies of points in projective space and applications (Orecchia; Ferruccio et al., eds.), Zero-Dimensional Schemes, Proceedings of the International Conference Held in Ravello, Italy, June 8–13, 1992, de Gruyter, Berlin, 1994, pp. 145-170

[4] W. Fulton Hurwitz schemes and irreducibility of moduli of algebraic curves, Ann. of Math., Volume 90 (1969), pp. 541-575

[5] M. Green Koszul cohomology and the geometry of projective varieties, J. Differential Geom., Volume 19 (1984), pp. 125-171 (With an Appendix by M. Green and R. Lazarsfeld)

[6] M. Green Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom., Volume 20 (1984), pp. 279-289

[7] M. Green; R. Lazarsfeld On the projective normality of complete linear series on an algebraic curve, Invent. Math., Volume 83 (1986), pp. 73-90

[8] M. Teixidor i Bigas Green's conjecture for the generic r-gonal curve of genus g⩾3r−7, Duke Math. J., Volume 111 (2002), pp. 363-404

[9] C. Voisin Green's generic syzygy conjecture for curves of even genus lying on a K3 surface, J. European Math. Soc., Volume 4 (2002), pp. 363-404

[10] C. Voisin, Green's canonical syzygy conjecture for generic curves of odd genus, Preprint, | arXiv

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