  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.
