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

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.