  Let X be a smooth, connected, dimension n, quasi-projective variety imbedded
in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme
H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with
multiplicities k_i at points x_i (possibly coinciding). The expected dimension
of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points
where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that
the lines carrying this locus do not fill up \PP_N
