La conjecture de Green générique  [ The generic Green conjecture ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 924, p. 1-14

A smooth projective curve C of genus g, non hyperelliptic, admits a canonical embedding in a projective space g-1 . It is classical that the graded ideal I C of equations of C in 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 g-1 . Un résultat classique affirme que l’idéal gradué I C des équations de C dans 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
Keywords: Green conjecture, syzygies, Clifford index, p-gonal curves
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     author = {Beauville, Arnaud},
     title = {La conjecture de Green g\'en\'erique},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:924},
     pages = {1-14},
     zbl = {1080.14041},
     mrnumber = {2167199},
     language = {fr},
     url = {http://www.numdam.org/item/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://www.numdam.org/item/SB_2003-2004__46__1_0/

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