We study graded algebras associated to 0-dimensionnal Gorenstein ideals I ⊂ ℂ[X_{0}, ⋯, X_{r}] with fixed socle degree and such that I contains a complete intersection of fixed multidegree. We conjecture that among such algebras those minimising the Hilbert function are some complete intersections. We prove an asympotic version of this conjecture.

Our result is an important case of a conjecture of Green, Eisenbud and Harris. In Hodge theory, it enables us to describe the biggest component of the Noether-Lefschetz locus of projective hypersurfaces of very high degree, in any dimension.