La conjecture de Green générique

Beauville, Arnaud

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 924, pp. 1-14.

A smooth projective curve $C$ of genus $g$, non hyperelliptic, admits a canonical embedding in a projective space ${\mathbb{P}}^{g-1}$. It is classical that the graded ideal $I_C$ of equations of $C$ in ${\mathbb{P}}^{g-1}$ is spanned by its elements of degree $2$, unless $C$ carries some very particular linear systems. Twenty years ago Mark Green proposed a far-reaching generalization, describing the minimal resolution of $I_C$ in terms of the existence of certain linear systems on $C$. Claire Voisin proved recently certain cases of the conjecture, notably the case of generic curves. We will try to explain the ideas which enter into this difficult proof.

Une courbe $C$ projective et lisse de genre $g$, non hyperelliptique, admet un plongement canonique dans un espace projectif ${\mathbb{P}}^{g-1}$. Un résultat classique affirme que l’idéal gradué $I_C$ des équations de $C$ dans ${\mathbb{P}}^{g-1}$ est engendré par ses éléments de degré $2$, sauf si $C$ admet certains systèmes linéaires très particuliers. Mark Green en a proposé il y a vingt ans une vaste généralisation, qui décrit la résolution minimale de $I_C$ en fonction de l’existence de systèmes linéaires spéciaux sur $C$. Claire Voisin vient de la démontrer dans un certain nombre de cas, et en particulier pour les courbes générales de genre donné. On essaiera d’expliquer les idées qui sous-tendent cette démonstration difficile.

Classification: 14H51, 13D02
Mot clés : conjecture de Green, syzygies, indice de Clifford, courbes $p$-gonales
Keywords: Green conjecture, syzygies, Clifford index, $p$-gonal curves

[B] E. Ballico - “On the Clifford index of algebraic curves”, Proc. Amer. Math. Soc. 97 (1986), p. 217-218. | MR | Zbl

[E] L. Ein - “A remark on the syzygies of the generic canonical curves”, J. Differential Geom. 26 (1987), p. 361-365. | MR | Zbl

[ELMS] D. Eisenbud, H. Lange, G. Martens & F.-O. Schreyer - “The Clifford dimension of a projective curve”, Compositio Math. 72 (1989), p. 173-204. | Numdam | MR | Zbl

[G] M. Green - “Koszul cohomology and the geometry of projective varieties”, J. Differential Geom. 19 (1984), p. 125-171. | MR | Zbl

[H-R] A. Hirschowitz & S. Ramanan - “New evidence for Green's conjecture on syzygies of canonical curves” 31 (1998), p. 145-152. | Numdam | MR | Zbl

[L] R. Lazarsfeld - “Brill-Noether-Petri without degenerations”, J. Differential Geom. 23 (1986), p. 299-307. | MR | Zbl

[Lo] F. Loose - “On the graded Betti numbers of plane algebraic curves”, Manuscripta Math. 64 (1989), p. 503-514. | MR | Zbl

[M] H. Martens - “Varieties of special divisors on a curve, II”, J. reine angew. Math. 233 (1968), p. 89-100. | MR | Zbl

[N] M. Noether - “Über die invariante Darstellung algebraischer Funktionen”, Math. Ann. 17 (1880), p. 263-284. | JFM | MR

[P-R] K. Paranjape & S. Ramanan - “On the canonical ring of a curve”, in Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, p. 503-516. | MR | Zbl

[P] K. Petri - “Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen”, Math. Ann. 88 (1923), p. 242-289. | JFM | MR

[S-D] B. Saint-Donat - “On Petri's analysis of the linear system of quadrics through a canonical curve”, Math. Ann. 206 (1973), p. 157-175. | MR | Zbl

[S1] F.-O. Schreyer - “Syzygies of canonical curves and special linear series”, Math. Ann. 275 (1986), p. 105-137. | MR | Zbl

[S2] -, “Green’s conjecture for general $p$-gonal curves of large genus”, in Algebraic curves and projective geometry (Trento, 1988), Lect. Notes in Math., vol. 1389, Springer, Berlin, 1989, p. 254-260. | MR | Zbl

[S3] -, “A standard basis approach to syzygies of canonical curves”, J. reine angew. Math. 421 (1991), p. 83-123. | MR | Zbl

[T] M. Teixidor I Bigas -Duke Math. J. 111 (2002), p. 195-222. | MR | Zbl

[V1] C. Voisin - “Courbes tétragonales et cohomologie de Koszul”, J. reine angew. Math. 387 (1988), p. 111-121. | MR | Zbl

[V2] -, “Green’s generic syzygy conjecture for curves of even genus lying on a $K3$ surface”, J. Eur. Math. Soc. 4 (2002), p. 363-404. | MR | Zbl

[V3] -, “Green's canonical syzygy conjecture for generic curves of odd genus”, Compositio Math. (à paraître), preprint arXiv : math.AG/0301359.