A DEFORMATION THEORY APPROACH TO LINEAR SYSTEMS WITH GENERAL TRIPLE POINTS

Let p1,..., pn be general points in the complex projective plane P2 andlet m1,...,mn be positive integers. We let Ld(pm11,..., pmnn) be the linearsystem of plane curves of degree d having multiplicity at least mi at the pointpi, i = 1,..., n. If mi = 1 we suppress the superscript mi for pi in Ld(pm11,..., pmnn).Let π: S → P2 be the blow-up of P2 at the points p1,..., pn. Let L bea line bundle on S, or, by abusing notation, the corresponding complete linearsystem. One de�nes the virtual dimension of L to be: ν(L): = χ(L) − 1 = L · (L − KS)2 where KS is the canonical class on S.If C is any divisor on S, we similarly de�ne ν(C): = χ(OS(C)) − 1. TheRiemann-Roch Theorem says that if L is effective, the