Bounding cohomology on a smooth projective surface

TL;DR

This paper investigates a conjecture relating the first and zeroth cohomology groups of line bundles on smooth projective surfaces, proving it for certain surfaces with Picard number 2.

Contribution

It establishes the conjecture for smooth projective surfaces with Picard number 2, providing new bounds on cohomology dimensions.

Findings

The conjecture holds for some surfaces with Picard number 2.

A positive constant c_X bounds h^1 in terms of h^0 for these surfaces.

The result advances understanding of cohomology behavior on algebraic surfaces.

Abstract

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