Detaching Embedded Points

Suppose that closed subschemes X ⊂ Y ⊂ P differ at finitely many points: when is Y a flat specialization of X union isolated points? Our main result says that this holds if X is a local complete intersection of codimension two and the multiplicity of each embedded point of Y is at most three. We show by example that no hypothesis can be weakened: the conclusion fails for (a) embedded points of multiplicity greater than three, (b) local complete intersections X of codimension greater than two and (c) non-local complete intersections of codimension two. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus and show the smoothness of the Hilbert component whose general member is a plane curve union a point in P.