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 $\mathbb{P}^2$: $I_n=(x(y^{n-zn),y(zn-xn),z(xn-yn))} \subset R=\mathbb{C}[x,y,z]$. 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.