  In this revised form, the proof of the principal lemma has been simplified
and the main theorem has been extended to all characteristics for those
varieties which are smooth in codimension one.
  This principal theorem essentially says the following : given an ample line
bundle O(1) on a projective variety X and a fixed upper bound M on the
multiplicities, there exists a lower bound D such that any generic union of
multiple points of multiplicity at most M imposes independent conditions on the
sections of O(d) for d>D. Here a multiple point is the closed subscheme defined
by a power m of the ideal of a smooth point in X and m is its multiplicity.
