  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 $X$. When the Picard
number $\rho(X)=2$, we prove that if either the Kodaira dimension $\kappa(X)=1$
and $X$ has a negative curve or $X$ has two negative curves, then this
conjecture holds for $X$.
