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 P^3.