Linear systems on generic $K3$ surfaces

Let S be a generic K3 surface (i.e. Pic S is generated by one element H), let P1, . . . , Pr be general points on S and fix r natural numbers m1, . . . mr. By L we denote the linear system of curves on S in |dH| (d > 0) which have multiplicities at least mi at Pi for all i = 1, . . . , r. Define the virtual dimension v of L as dim |dH| − ∑ rimi(mi + 1)/2 and its expected dimension by e = max{v,−1}. Obviously, the dimension l of L is allways bigger or equal to e, and if l > e we say that the system is special.