The Asymptotics of Symbolic Generic Initial Systems of Six Points in P2

TL;DR

This paper investigates the asymptotic behavior of symbolic generic initial systems for ideals of six points in the projective plane, providing a detailed description of their limiting shapes across all configurations.

Contribution

It characterizes the limiting shape of symbolic generic initial systems for six points in P2, covering all possible point configurations, and advances understanding of asymptotic properties in algebraic geometry.

Findings

Describes the limiting shape for each configuration type

Provides a comprehensive classification for six points in P2

Advances understanding of symbolic powers and initial ideals

Abstract

Consider the ideal I in K[x,y,z] corresponding to six points of P2. We study the limiting behaviour of the symbolic generic initial system, of I obtained by taking the reverse lexicographic generic initial ideals of the symbolic powers of I. The main result of this paper is a theorem describing the limiting shape of the symbolic generic initial system for each of the eleven possible configuration types of six points.