Bounding cohomology on a smooth projective surface

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 $X$. We show that the conjecture is true for some smooth projective surfaces with Picard number 2.

11 pages