An infinitesimal approach to a conjecture of Eisenbud and Harris

Abstract

Eisenbud and Harris conjectured in 1982 that an algebraic curve of high genus lies on a surface of low degree (which they proved for curves of very large degree). They observed constraints on the Hilbert function of a general hyperplane section which imply that the hyperplane section lies on a curve of low degree. We investigate this situation under deformation. Given a set of sufficiently many points (as postulated by the conjecture) on a linearly normal curve, we show that if the number of conditions on quadrics remains constant, then for every deformation of the points the curve deforms along with them.