A conjectural generalization of the Briançon-Iarrobino Conjecture

2025/06/21 by Ascott, Alexia, Rezaee, Fatemeh, Zhou, Zhichen
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2506.17704

Abstract

We conjecturally generalize the Conjecture by Briançon and Iarrobino on the most singular points of the Hilbert scheme of a tetrahedral number of points in \BA³ by introducing the notion of 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 \citeRezaee-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.

Citations

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• A sign that used to annoy me, and still does
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Related

• A proof of the Briançon-Iarrobino Conjecture in three dimensions
• Conjectural criteria for the most singular points of the Hilbert schemes of points
• On singular Hilbert schemes of points: Local structures and tautological sheaves
• The Hilbert scheme of points on a threefold, I
• Segments and Hilbert schemes of points