We study the problem of classifying the irreducible projective varieties X
of dimension n≥2 in PN which contain an algebraic family \Cal F
of dimension h+1 (h<n) of subvarieties Y of dimension n−h, each one
contained in a PN−h−1. We prove that one of the following happens:
(i) there exists an integer r, r<N−n such that X is contained in a
variety Vr​ of dimension at most N−r containing a family of dimension h+1
of subvarieties of dimension N−h−r, each one contained in a linear space of
dimension N−h−1; (ii) The degree of Y is bounded by a function of h and
N−n (in this case X is called of isolated type). Successively we study some
special cases; in particular we give a complete classification of surfaces in
P5 containing a family of dimension 2 of curves of P3.Comment: 19 pages, AMS-TeX 2.