Non-reducedness of the Hilbert schemes of few points

TL;DR

This paper proves that the Hilbert scheme of 13 points on six-dimensional affine space is non-reduced, using advanced geometric and algebraic tools without computer assistance, revealing complex internal structures.

Contribution

It introduces a novel approach combining Bialynicki-Birula decomposition, apolarity, and obstruction theories to establish non-reducedness without computational methods.

Findings

The Hilbert scheme of 13 points on ^6 is non-reduced.

The proof reveals a fractal-like internal structure.

The method avoids computer calculations, offering a new theoretical framework.

Abstract

We use generalised Bialynicki-Birula decomposition, apolarity and obstruction theories to prove non-reducedness of the Hilbert scheme of $13$ points on $\mathbb{A}^6$. Our argument doesn't involve computer calculations and gives an example of a fractal-like structure on this Hilbert scheme.