The Initial Degree of Symbolic Powers of Fermat-like Ideals of Planes and Lines Arrangements

TL;DR

This paper explicitly computes the initial degrees of symbolic powers of Fermat-like line arrangements in projective space, verifies several conjectures, and calculates key invariants related to these ideals.

Contribution

It provides explicit degree computations for symbolic powers of Fermat-like ideals and verifies several important conjectures in algebraic geometry.

Findings

Explicit degrees of symbolic powers for Fermat-like line arrangements

Verification of Chudnovsky's, Demailly's, and Harbourne-Huneke conjectures

Calculation of Waldschmidt constant and resurgence number

Abstract

We explicitly compute the least degree of generators of all symbolic powers of the defining ideal of Fermat-like configuration of lines in $\mathbb{P}^3_\mathbb{C}$, except for the second symbolic powers, where we provide bounds for them. We will also explicitly compute those numbers for ideal determining the singular locus of the arrangement of lines given by the pseudoreflection group $A_3$. As direct applications, we verify Chudnovsky's(-like) Conjecture, Demailly's(-like) Conjecture and Harbourne-Huneke Containment problem as well as calculate the Waldschmidt constant and (asymptotic) resurgence number.