A monotonicity conjecture for the local maximal singularity of the Hilbert scheme of points
Alexia Ascott, Fatemeh Rezaee, Zhichen Zhou
TL;DR
This work investigates maximal singularities in the Hilbert scheme of points $\mathrm{Hilb}^l(\mathbb{A}^N)$, aiming to extend understanding beyond tetrahedral $l$ by focusing on tangent-space dimensions at points corresponding to Borel-fixed ideals. It introduces the notion of locally-maximal (or $m_1$-maximal) singularities, defined via the smallest pure exponent $m_1$ in a Borel-fixed ideal $I=(x_1^{m_1},x_2^{m_2},\ldots,x_N^{m_N},\text{mixed terms})$, and denotes $T(I)=\dim \mathrm{Hom}(I,R/I)$ with $T_{\max,m_1}(l)$ the maximum over colength $l$ for fixed $m_1$. The main contribution is a monotonicity conjecture asserting that $T_{\max,m_1}(l)$ increases with $m_1$, together with a conjectural sufficient condition for the necessary condition; the work also explores a pattern for locally-maximal singularities that underpins this conjecture. Computational evidence for $N=3$ with $10\le l\le 35$ supports the proposed monotonicity, and Macaulay2 code is provided in the Appendix to facilitate further verification and potential proof strategies.
Abstract
The Briançon-Iarrobino conjecture predicts the maximum singularity of the Hilbert scheme of a tetrahedral number of points. As for the maximal singularities of the Hilbert scheme of a non-tetrahedral number of points, the second named author gave some separate conjectural necessary and sufficient conditions. In this paper, we provide a conjectural sufficient condition for the necessary condition, and propose a monotonicity conjecture which predicts that for a fixed colength $l$, the maximal dimension of the tangent space over all the Borel-fixed ideals of colength $l$ is increasing with respect to the smallest pure exponent of the ideal.