The asymptotic behaviour of symbolic generic initial systems of points in general position

TL;DR

This paper investigates the asymptotic behavior of symbolic generic initial systems of points in general position in projective space, describing their limiting shape and connections to Hilbert functions, under the SHGH Conjecture.

Contribution

It explicitly characterizes the limiting shape of symbolic generic initial systems for points in general position, linking algebraic and geometric properties.

Findings

Explicit description of the limiting shape using SHGH Conjecture

Connection between generic initial systems and Hilbert functions

Insights into the asymptotic behavior of symbolic powers

Abstract

Consider the ideal I corresponding to r points in P^2. We study the symbolic generic initial system of I, formed by taking the generic initial ideals of the symbolic powers of I, and its asymptotic behaviour. In particular, we describe the limiting shape of this system explicitly when the points lie in general position using the SHGH Conjecture for more than eight points. The symbolic generic initial system and its limiting shape reflects information about the Hilbert functions of uniform fat point ideals.