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

TL;DR

This paper investigates the asymptotic behavior of symbolic generic initial systems for a special class of point configurations in P2, revealing their limiting shape and demonstrating componentwise linearity of related ideals.

Contribution

It introduces the analysis of symbolic generic initial systems for almost linear point configurations in P2 and establishes their limiting shape and componentwise linearity.

Findings

Describes the limiting shape of the symbolic generic initial system.

Shows infinitely many uniform fat point ideals are componentwise linear.

Provides new insights into the structure of ideals from point configurations in P2.

Abstract

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.