Maps of linear systems on blow-ups of the projective plane

Let X be the blow-up of P2 at n points p1,…,pn in linearly general position, let F be a numerically effective divisor on X, and let L be the total transform on X of a line on P2. The natural multiplication map image is shown to be surjective for a particular subset of numerically effective divisors. This result implicitly determines minimal free resolutions of ideals defining fat point subschemes supported at six or fewer general points of P2. All work is over an algebraically closed field k.