Bounding cohomology on a smooth projective surface with Picard number 2

Abstract

The following conjecture arose out of discussions between B. Harbourne, J. Roé, 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 $\ddot{I}(X)=2$, we prove that if either the Kodaira dimension $\hat{I}º(X)=1$ and $X$ has a negative curve or $X$ has two negative curves, then this conjecture holds for $X$.

6 pages, to appear in Communications in Algebra, minor revision