Degeneration of linear systems through fat points on $K3$ surfaces

In this paper we introduce a technique to degenerate $K3$surfaces and linear systems through fat points in general position on $K3$surfaces. Using this degeneration we show that on generic $K3$ surfaces it is enough to prove that linear systems with one fat point are non-special in order to obtain the non-speciality of homogeneous linear systems through $n = 4^u9^w$fat points in general position. Moreover, we use this degeneration to obtain a result for homogeneous linear systems through $n = 4^u9^w$ fat points in general position on a general quartic surface in $\mathbb{P}^3$.