An algorithm for fat points on P-2

Let F be a divisor on the blow-up X of P-2 at r general points p(1),..., p( r) and let L be the total transform of a line on P-2. An approach is presen ted for reducing the computation of the dimension of the cokernel of the na tural map mu(F): Gamma(O-X(F)) x Gamma(O-X(L)) --> Gamma(O-X(F) x O-X(L)) t o the case that F is ample. As an application, a formula for the dimension of the cokernel of mu(F) is obtained when r = 7, completely solving the pro blem of determining the modules in minimal free resolutions of fat point su bschemes m(1)p(1) +...+ m(7)p(7) subset of P-2. All results hold for an arb itrary algebraically dosed ground field k.