The Symbolic Generic Initial System of Points on an Irreducible Conic

Abstract

In this note we study the limiting behaviour of the symbolic generic initial system {gin(I (m) )} of an ideal I ⊆ K[x, y, z] corresponding to an arrangement of r points of P 2 lying on an irreducible conic. In particular, we show that the limiting shape of this system is the subset of R 2 ≥0 such consisting of all points above the line through (min{ r 2 , 2}, 0) and (0, max{ r 2 , 2}).

The asymptotic behaviour of symbolic generic initial systems of generic points

“…Suppose that I is the ideal corresponding to one of the following subsets of P 2 : a point configuration of at most six points; a point configuration arising from a complete intersection; a generic set of points; points on an irreducible conic; a point configuration where all but one point lies on a line; a point configuration where all but two points lies on a line and no other line passes through three points; or a star point configuration. Then the number of line segments forming the boundary of the limiting shape of {gin(I (m) )} m is equal to the number of distinct incidence types of the points in the corresponding point configuration ([May12a], [May13a], [May13b], [May12c]).…”