On Bounding the Number of Generators for Fat Point Ideals on the Projective Plane

Let X be the surface obtained by blowing up general points p1, …,pn of the projective plane over an algebraically closed ground field k, and let L be the pullback to X of a line on the plane. If C is a rational curve on X with C • L = d, then for every t there is a natural map Γ(C, C(t)) Γ(X, X(L)) → Γ(C, C(t + d)) given by multiplication on simple tensors. The ranks of such maps are determined as a function of t, d, and m, where m is the largest multiplicity of C at any of the points pi. If I is the ideal defining the fat point subscheme Z = m1p1 + ••• + mnpn P2, and α is the least degree in which I has generators, then the ranks of the maps Γ(C, C(t)) Γ(X, X(L)) → Γ(C, C(t + d)) can be used for bounding the number of generators of I in degrees t > α + 1.