A smooth projective curve of genus , non hyperelliptic, admits a canonical embedding in a projective space . It is classical that the graded ideal of equations of in is spanned by its elements of degree , unless carries some very particular linear systems. Twenty years ago Mark Green proposed a far-reaching generalization, describing the minimal resolution of in terms of the existence of certain linear systems on . 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 projective et lisse de genre , non hyperelliptique, admet un plongement canonique dans un espace projectif . Un résultat classique affirme que l’idéal gradué des équations de dans est engendré par ses éléments de degré , sauf si 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 en fonction de l’existence de systèmes linéaires spéciaux sur . 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.
Mot clés : conjecture de Green, syzygies, indice de Clifford, courbes $p$-gonales
Keywords: Green conjecture, syzygies, Clifford index, $p$-gonal curves
@incollection{SB_2003-2004__46__1_0, author = {Beauville, Arnaud}, title = {La conjecture de {Green} g\'en\'erique}, booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937}, series = {Ast\'erisque}, note = {talk:924}, pages = {1--14}, publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France}, address = {Paris}, number = {299}, year = {2005}, mrnumber = {2167199}, zbl = {1080.14041}, language = {fr}, url = {http://archive.numdam.org/item/SB_2003-2004__46__1_0/} }
TY - CHAP AU - Beauville, Arnaud TI - La conjecture de Green générique BT - Séminaire Bourbaki : volume 2003/2004, exposés 924-937 AU - Collectif T3 - Astérisque N1 - talk:924 PY - 2005 SP - 1 EP - 14 IS - 299 PB - Association des amis de Nicolas Bourbaki, Société mathématique de France PP - Paris UR - http://archive.numdam.org/item/SB_2003-2004__46__1_0/ LA - fr ID - SB_2003-2004__46__1_0 ER -
%0 Book Section %A Beauville, Arnaud %T La conjecture de Green générique %B Séminaire Bourbaki : volume 2003/2004, exposés 924-937 %A Collectif %S Astérisque %Z talk:924 %D 2005 %P 1-14 %N 299 %I Association des amis de Nicolas Bourbaki, Société mathématique de France %C Paris %U http://archive.numdam.org/item/SB_2003-2004__46__1_0/ %G fr %F SB_2003-2004__46__1_0
Beauville, Arnaud. La conjecture de Green générique, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 924, pp. 1-14. http://archive.numdam.org/item/SB_2003-2004__46__1_0/
[B] “On the Clifford index of algebraic curves”, Proc. Amer. Math. Soc. 97 (1986), p. 217-218. | MR | Zbl
-[E] “A remark on the syzygies of the generic canonical curves”, J. Differential Geom. 26 (1987), p. 361-365. | MR | Zbl
-[ELMS] “The Clifford dimension of a projective curve”, Compositio Math. 72 (1989), p. 173-204. | Numdam | MR | Zbl
, , & -[G] “Koszul cohomology and the geometry of projective varieties”, J. Differential Geom. 19 (1984), p. 125-171. | MR | Zbl
-[H-R] “New evidence for Green's conjecture on syzygies of canonical curves” 31 (1998), p. 145-152. | Numdam | MR | Zbl
& -[L] “Brill-Noether-Petri without degenerations”, J. Differential Geom. 23 (1986), p. 299-307. | MR | Zbl
-[Lo] “On the graded Betti numbers of plane algebraic curves”, Manuscripta Math. 64 (1989), p. 503-514. | MR | Zbl
-[M] “Varieties of special divisors on a curve, II”, J. reine angew. Math. 233 (1968), p. 89-100. | MR | Zbl
-[N] “Über die invariante Darstellung algebraischer Funktionen”, Math. Ann. 17 (1880), p. 263-284. | JFM | MR
-[P-R] “On the canonical ring of a curve”, in Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, p. 503-516. | MR | Zbl
& -[P] “Über die invariante Darstellung algebraischer Funktionen einer Veränderlichen”, Math. Ann. 88 (1923), p. 242-289. | JFM | MR
-[S-D] “On Petri's analysis of the linear system of quadrics through a canonical curve”, Math. Ann. 206 (1973), p. 157-175. | MR | Zbl
-[S1] “Syzygies of canonical curves and special linear series”, Math. Ann. 275 (1986), p. 105-137. | MR | Zbl
-[S2] -, “Green’s conjecture for general -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] I Bigas -Duke Math. J. 111 (2002), p. 195-222. | MR | Zbl
[V1] “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 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.