La conjecture de Green générique

Beauville, Arnaud

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

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.

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.

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

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