A few remarks on the lifting theorem.

We start with a projective variety X in Pr and a family W of projective subvarieties of Pr, parametrized by the space B, such that for any tin B the corresponding fibre Wt of W is contained in some h-plane Lt and Wt contains XnLt; furthermore, as t ranges in B, let the Lt's fill an open dense subset of the corresponding Grassmannian. We give conditions on the degrees of X, Wt which implies that the varieties Wt glue together giving a variety W (containing X) such that Wt=WnLt for all t. Our conditions, whose proves are based on the classical differential theory of "foci" introduced by C.Segre, generalize the well-known theorems of Laudal and Gruson - Peskine for the case X = curve in P3.