  For an irreducible projective variety X, we study the family of h-planes
contained in the secant variety Sec_k(X), for 0<h<k. These families have an
expected dimension and we study varieties for which the expected dimension is
not attained; for these varieties, making general consecutive projections to
lower dimensional spaces, we do not get the expected singularities. In
particular, we examine the family G of lines sitting in 3-secant planes to a
surface S. We show that the actual dimension of G is equal to the expected
dimension unless S is a cone or a rational normal scroll of degree 4 in P^5.
