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La conjecture de Green générique

Arnaud Beauville

Séminaire Bourbaki

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...

Orthogonal bundles on curves and theta functions

Arnaud Beauville — 2006

Annales de l’institut Fourier

Let be the moduli space of principal SO r -bundles on a curve C , and the determinant bundle on . We define an isomorphism of H 0 ( , ) onto the dual of the space of r -th order theta functions on the Jacobian of C . This isomorphism identifies the rational map | | * defined by the linear system | | with the map | r Θ | which associates to a quadratic bundle ( E , q ) the theta divisor Θ E . The two components + and - of are mapped into the subspaces of even and odd theta functions respectively. Finally we discuss the analogous...

Some surfaces with maximal Picard number

Arnaud Beauville — 2014

Journal de l’École polytechnique — Mathématiques

For a smooth complex projective variety, the rank ρ of the Néron-Severi group is bounded by the Hodge number h 1 , 1 . Varieties with ρ = h 1 , 1 have interesting properties, but are rather sparse, particularly in dimension 2 . We discuss in this note a number of examples, in particular those constructed from curves with special Jacobians.

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