The Symbolic Generic Initial System of Points on an Irreducible Conic

In this note we study the limiting behaviour of the symbolic generic initial system of an ideal I in K[x,y,z] corresponding to an arrangement of r points of P2 lying on an irreducible conic. In particular, we show that the limiting shape of this system is the subset of R2 such consisting of all points above the line through (2,0) and (0, r/2) when r is greater than or equal to four.