Curves on threefolds : On a conjecture of Griffiths-Harris

Mardi 4 novembre 2008 16:00-17:00 - Ravindra G.V. - Indian Institute of Science

Résumé : We will sketch a proof of the following characterisation of complete intersection curves on a general smooth projective hypersurface of dimension three and degree at least six : « a curve in such a hypersurface is a complete intersection if and only if it is arithmetically Gorenstein (i.e. it is the zero locus of a non-zero section of a rank two bundle with vanishing intermediate cohomology). » Apart from the obvious motivation for such a theorem, we shall show
1) how this theorem can be thought of as a generalisation of the classical Noether-Lefschetz theorem for curves on two dimensional hypersurfaces. A stronger generalisation was conjectured by Griffiths and Harris which is known to be false thanks to Voisin.
2) that this implies the following interesting fact : a « general » homogeneous polynomial in five variables of degree at least six cannot be obtained as the Pfaffian (square root of the determinant) of an even sized « minimal » skew-symmetric matrix with homogeneous polynomial entries.
3) Relate it to Horrocks’ criteria for a vector bundle on projective space to be split, and finally
4) show that it can be viewed as a verification of a (strengthening of a) conjecture of Buchweitz-Greuel-Schreyer in commutative algebra and algebraic geometry.
5) Time permitting, we shall show how Voisin’s example is a special case of a large class of counter-examples to this conjecture.

Lieu : bât. 425 - 113-115

Curves on threefolds : On a conjecture of Griffiths-Harris  Version PDF