A note on bounding cohomology on a smooth projective surface.

A question 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 $X$. We first note that this conjecture is true for such $X$ with the Picard number $\rho(X)=1$. And then when $\rho(X)=2$, we prove that if the Kodaira dimension $\kappa(X)=-\infty$ and $X$ has a negative curve, then this conjecture holds for $X$.