Direct images of vector bundles on rational surfaces

The direct images of a wide class of vector bundles over general blowups of the projective plane are characterized. Direct images are taken under a rational fibration map to the projective line. These direct images are then direct sums of line bundles, and their cohomology is isomorphic to that of the original bundles. Explicit bounds for the degrees of the component line bundles as well as complete computations are obtained. It is also shown how the datum of the second difference of a suitable arithmetic function determines a direct image.