  In this note we address the following kind of question: let X be a smooth,
irreducible, projective surface and D a divisor on X$satisfying some sort of
positivity hypothesis, then is there some multiple of D depending only on X
which is effective or movable? We describe some examples, discuss some
conjectures and prove some results that suggest that the answer should in
general be negative, unless one puts some really strong hypotheses either on D
or on X.
