Bounding cohomology on a smooth projective surface with Picard number 2

TL;DR

This paper investigates a conjecture relating the first and zeroth cohomology groups of line bundles on smooth projective surfaces with Picard number 2, proving it in specific cases involving negative curves and Kodaira dimension.

Contribution

The paper proves the conjecture for surfaces with Picard number 2 when certain conditions on negative curves and Kodaira dimension are met, advancing understanding of cohomology bounds.

Findings

The conjecture holds if the surface has a negative curve and Kodaira dimension 1.

The conjecture holds if the surface has two negative curves.

Results apply specifically to surfaces with Picard number 2.

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 $X$. When the Picard number $\rho(X)=2$, we prove that if either the Kodaira dimension $\kappa(X)=1$ and $X$ has a negative curve or $X$ has two negative curves, then this conjecture holds for $X$.