INITIAL DEGREE OF SYMBOLIC POWERS OF IDEALS OF FERMAT CONFIGURATIONS OF POINTS

Abstract

Let $n\ge 2$ be an integer and consider the defining ideal of the Fermat configuration of points in ${\mathrm{\mathbb{P}}}^2$: $I_n=\left(x(y^n-z^n),y(z^n-x^n),z(x^n-y^n)\right)\subset R=ℂ[x,y,z]$. We explicitly compute the least degree of generators (the initial degree) of its symbolic powers in all unknown cases. As direct applications, we verify Chudnovsky’s conjecture, Demailly’s conjecture and the Harbourne–Huneke containment problem, as well as calculate the Waldschmidt constant and (asymptotic) resurgence number.