# A proof of the Brian\c{c}on-Iarrobino Conjecture in three dimensions

## TL;DR

This paper proves the Briançon-Iarrobino Conjecture for three-dimensional Hilbert schemes at tetrahedral numbers, confirming the maximum singularity condition and advancing understanding of singularities in algebraic geometry.

## Contribution

It provides a proof of the conjecture for tetrahedral numbers in three dimensions, refining previous work and establishing necessary conditions for maximal singularities.

## Key findings

- Confirmed the conjecture for tetrahedral numbers in 3D
- Established necessary conditions for maximal singularities
- Extended results to certain non-tetrahedral cases in follow-up work

## Abstract

We resolve the 1978 Brian\c{c}on-Iarrobino Conjecture regarding the maximum singularity of $\mathcal{H}=\mathrm{Hilb}^{l}(\mathbb{A}^3)$, where $l$ is a tetrahedral number, by refining the work of Ramkumar-Sammartano in \cite{Ramkumar-Sammartano}. This also immediately implies the conjectural necessary condition for a point of $\mathcal{H}$ to have the maximal singularity, suggested by the second-named author in \cite{Rezaee-23-Conjectures}. In a sequel to this article, \cite{Mackenzie-Rezaee2}, we prove a generalized version of this conjecture for certain non-tetrahedral $l$, via proving the conjectural necessary condition.