A conjectural generalization of the Briançon-Iarrobino Conjecture

Abstract: We conjecturally generalize the Conjecture by Brian\c{c}on and Iarrobino on the most singular points of the Hilbert scheme of a tetrahedral number of points in $\BA<sup>3$ by introducing the notion of \textit{locally-maximal singularity}. The formulation of this conjecture is based on the shape of the ideals via a picturesque pattern. It also generalizes the conjectural necessary condition for the most singular points, suggested by the second-named author in \cite{Rezaee-23-Conjectures}, restricted to a tetrahedral number of points in 3D. These conjectures and those in the prequel work aim to prove the original conjecture.