  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.
