  A conjecture of the bounding cohomology on a smooth projective surface $X$
asserts that there exists a positive constant $c_X$ such that $h^1(\mathcal
O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every prime divisor $C$ on $X$. When
the Picard number $\rho(X)=2$, we prove that if the Kodaira dimension
$\kappa(X)=-\infty$ and $X$ has a negative curve, then this conjecture holds
for $X$.
