  We conjecture and prove closed-form index expressions for the cohomology
dimensions of line bundles on del Pezzo and Hirzebruch surfaces. Further, for
all compact toric surfaces we provide a simple algorithm which allows
expression of any line bundle cohomology in terms of an index. These formulae
follow from general theorems we prove for a wider class of surfaces. In
particular, we construct a map that takes any effective line bundle to a nef
line bundle while preserving the zeroth cohomology dimension. For complex
surfaces, these results explain the appearance of piecewise polynomial
equations for cohomology and they are a first step towards understanding
similar formulae recently obtained for Calabi-Yau three-folds.
