We start with a projective variety X in $\mathbb {P}^r$ and a family $\mathsf {W}$ of projective subvarieties of $\mathbb {P}^r$, parametrized by the space B, such that for any $t \in $ B the corresponding fibre W$_t$ of $\mathsf {W}$ is contained in some $h$-plane L$_t$, and W$_t\,\supseteq $ X $\cap $ L$_t$ ; we assume that the L$_t$'s for variable $t$ fill an open dense subset of the corresponding Grassmannian. We give conditions on the degrees of X and W$_t$ which imply that the varieties W$_t$ glue together to give a variety W (containing X ) such that W$_t$ = W $\cap $ L$_t$ for all $t$. The proofs are based on the ical differential theory of “foci” introduced by C. Segre. Our results generaIize the theorems of Laudal and Gruson-Peskine, which deal with the case X is a curve in $\mathbb {P}^3$