Lifting Problem in codimension 2 and initial ideals

Let X be a codimension 2, locally Cohen-Macaulay, integral, projective variety of degree d in PN. We consider the problem of finding conditions on d, N and s such that any degree s hypersurface in PN−1 containing a general hyperplane section of X lifts to a hypersurface in PN containing X. We prove general and sharp bounds on the degree of X depending on both N and s and also on the number of independent hypersurfaces of degree s containing X, especially under the additional condition that the general plane section of X does not lie on any degree s − 1 curve