  The following conjecture arose out of discussions between B. Harbourne, J.
Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there
exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X
h^0(\mathcal O_X(C))$ for every prime divisor $C$ on $X$. We show that the
conjecture is true for some smooth projective surfaces with Picard number 2.
