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 p(1),..., p(n) o f 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 Gamma (C, O-C(t) ) circle times Gamma (X, O-X(L)) --> Gamma (C, O-C(t + d)) given by multipl ication on simple tensors. The ranks of such maps are determined as a funct ion of t, d, and m, where rn is the largest multiplicity of C at any of the points p(t). If I is the ideal defining the fat point subscheme Z = m(1)p( 1) + ... + m(n)p(n) subset of P-2, and a is the least degree in which I has generators, then the ranks of the maps T(C, O-C(t)) circle times Gamma (X, O-X(L)) --> Gamma (C, O-C(t + d)) can be used for bounding the number of g enerators of I in degrees t > alpha + 1. (C) 2001 Academic Press.