  Let X be a smooth projective surface over C and let L be an ample line bundle
on X. In this note, we show that, for all sufficiently large d, any number of
general double points on X imposes the expected number of conditions on the
linear system |L^d|. Equivalently, the space of d-plane sections of X singular
at any number of general points has the expected dimension. We conjecture that
the same holds for X of arbitrary dimension.
