Deformations of the connected sum of Gorenstein algebras

TL;DR

This work proves that the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{A}^n$ is non-reduced for $n>9$, by constructing explicit non-reduced points as apolar algebras of sums of general cubics. It develops an abstract Białynicki-Birula decomposition for good deformation functors, introducing filtered and pointed deformation frameworks, unions along a point, and connected sums to analyze local deformations. The authors further describe fractal structures on nested Hilbert schemes and show how these fractal features propagate to cactus schemes, providing new non-reduced phenomena beyond traditional components. The result yields concrete non-reduced points on ${\mathrm{Hilb}}^{Gor}_{d}(\mathbb{A}^n)$ (e.g., for $d=2m+2$, $n=m$, $m\ge 10$) and supplies a versatile deformation-theoretic toolkit for understanding the local geometry of the Gorenstein locus and related moduli spaces.

Abstract

We prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n>9$; we construct examples of non-reduced points that come from apolar algebras of the sum of general cubics. As a corollary, we get a non-reducedness result for the cactus scheme. We generalise the Białynicki-Birula decomposition to abstract deformation functors, providing a new method of studying deformation theory. Our construction gives us fractal structures on the nested Hilbert scheme.

Theorems & Definitions (157)

• Theorem 1.1: Corollary \ref{['main-result']}
• Corollary 1.2: Corollary \ref{['final-app']}
• Theorem : ABB_JJLS
• Theorem 1.3: Theorem \ref{['+obs']}
• Lemma 1.4: Lemma \ref{['frac-nonred']}
• Theorem 1.5: Theorem \ref{['fractal1']}
• Theorem 1.6: Theorem \ref{['step2-main']}
• Definition 2.1
• Example 2.2
• Definition 2.3