The Initial Degree of Symbolic Powers of Ideals of Fermat Configuration of Points

TL;DR

This paper explicitly computes the initial degrees of symbolic powers of ideals from Fermat point configurations, confirming several conjectures and calculating key invariants like the Waldschmidt constant and resurgence number.

Contribution

It provides explicit calculations of the initial degrees of symbolic powers for Fermat configuration ideals, verifying major conjectures and determining important algebraic invariants.

Findings

Confirmed Chudnovsky's Conjecture for these ideals

Verified Demailly's Conjecture and Harbourne-Huneke Containment

Computed Waldschmidt constant and resurgence number explicitly

Abstract

Let $n \ge 2$ be an integer and consider the defining ideal of the Fermat configuration of points in $\mathbb{P}^2$: $I_n=(x(y^n-z^n),y(z^n-x^n),z(x^n-y^n)) \subset R=\mathbb{C}[x,y,z]$. In this paper, we compute explicitly the least degree of generators of its symbolic powers in all unknown cases. As direct applications, we easily verify Chudnovsky's Conjecture, Demailly's Conjecture and Harbourne-Huneke Containment problem as well as calculating explicitly the Waldschmidt constant and (asymptotic) resurgence number.