  Consider an ideal I in K[x,y,z] corresponding to a point configuration in P2
where all but one of the points lies on a single line. In this paper we study
the symbolic generic initial system obtained by taking the reverse
lexicographic generic initial ideals of the corresponding uniform fat point
ideals. We describe the limiting shape of this system of ideals and, in proving
this result, demonstrate that infinitely many of the uniform fat point ideals
are componentwise linear.
