Bounding embedded singularities of Hilbert schemes of points on affine three space (Q6545241)

The author bounds how bad singularities of the Hilbert scheme \(\mathrm{Hilb}^n \mathbb C^3\) of \(n\) points on \(\mathbb C^3\) can be in the following sense. By work of \textit{B. Szendrői} [Geom. Topol. 12, No. 2, 1171--1202 (2008; Zbl 1143.14034)] and \textit{K. Behrend} et al. [Invent. Math. 192, No. 1, 111--160 (2013; Zbl 1267.14008)] on Donald-Thomas invariants, \(\mathrm{Hilb}^n \mathbb C^3\) can be realized as the critical locus \(d \tilde f = 0\) for a regular function \(\tilde f: \mathcal X \to \mathbb C\) with \(\mathcal X\) a smooth quasi-projective variety of dimension \(2n^2+n\). Therefore one can view \(\mathrm{Hilb}^n \mathbb C^3 \cong \mathcal Y \subset \mathcal X\) as a closed subscheme. The \textit{multiplicity} of a point \(\eta \in Y\) is defined by \(\mathrm{mult}_{\eta} \mathcal Y = \max\{q > 0 : I_{\mathcal Y,\eta} \subset \mathfrak m_{\mathcal X, \eta}^q\}\). Following the technique of \textit{M. Mustaţă} [J. Am. Math. Soc. 15, No. 3, 599--615 (2002; Zbl 0998.14009)], the author estimates the log canonical threshold \(\mathrm{lct}_{\eta} (\mathcal X, \mathcal Y)\) for arbitrary \(\eta \in \mathcal Y\) using jet schemes, then combines with the inequalities\N\[\N\frac{1}{\mathrm{mult}_{\eta} \mathcal Y} \leq \mathrm{lct}_{\eta} (\mathcal X, \mathcal Y) \leq \frac{\dim \mathcal X}{\mathrm{mult}_{\eta} \mathcal Y}\N\]\Nto bound \(q_n = \max\{\mathrm{mult}_{\eta} \mathcal Y, \eta \in \mathcal Y\}\) and concludes that \(q_n\) has at most linear growth in \(n\). In particular, the Hilbert schemes \(\mathrm{Hilb}^n \mathbb C^3\) do not have arbitrarily bad singularities in the sense of \textit{R. Vakil} [Invent. Math. 164, No. 3, 569--590 (2006; Zbl 1095.14006)]. By contrast, \textit{J. Jelisiejew} has shown that \(\mathrm{Hilb}^n \mathbb C^{16}\) does have arbitrarily bad singularies up to retraction [Invent. Math. 220, No. 2, 581--610 (2020; Zbl 1473.14007)]. The proof uses the Atiyah-Drinfeld-Hitchin-Manin description relating irreducible components of \(\mathrm{Hilb}^n \mathbb C^d\) with those of the scheme \(C_{d,n}\) parametrizing \(d\) commuting \(n \times n\) matrices (see \textit{A. A. Henni} and \textit{M. Jardim} [Adv. Geom. 18, No. 4, 467--482 (2018; Zbl 1403.14015)] and \textit{J. Jelisiejew} and \textit{K. Šivic} [J. Reine Angew. Math. 788, 129--187 (2022; Zbl 1494.14007)]).