Deformations of the connected sum of Gorenstein algebras

TL;DR

This paper demonstrates non-reducedness in the Gorenstein locus of the Hilbert scheme of points on affine space for dimensions 12 and above, introduces a new deformation method, and explores fractal structures in nested Hilbert schemes.

Contribution

It proves non-reducedness of the Gorenstein locus for high dimensions, constructs examples from apolar algebras of cubics, and generalizes Bia{ }ynicki-Birula decomposition to deformation functors.

Findings

Gorenstein locus is non-reduced for n ≥ 12

Constructed non-reduced points from apolar algebras of cubics

Established fractal structures on nested Hilbert schemes

Abstract

We prove that the Gorenstein locus of the Hilbert scheme of points on $\mathbb A^n$ is non-reduced for $n\geq 12$; 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 Bia{\l}ynicki-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.