The Symbolic Generic Initial System of Almost Linear Point Configurations in P2

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.