Grassmannians of secant varieties

For an irreducible projective variety X, we study the family of h-planes co ntained in the secant variety S-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 projectio ns to lower dimensional spaces, we do not get the expected singularities. I n particular, we examine the family G(1,2) of lines sitting in 3-secant pla nes to a surface S. We show that the actual dimension of G(1,2) is equal to the expected dimension unless S is a cone or a rational normal scroll of d egree 4 in P-5.