# Pathologies on the Hilbert scheme of points

## TL;DR

This paper demonstrates that the Hilbert scheme of points in higher-dimensional affine space exhibits non-reduced components and characteristic-dependent behavior, confirming a form of Vakil's Murphy's Law using advanced geometric techniques.

## Contribution

It proves the non-reducedness and characteristic-dependent components of the Hilbert scheme of points in higher dimensions, extending Vakil's Murphy's Law to this setting.

## Key findings

- Hilbert scheme of points is non-reduced in higher dimensions.
- Components of the scheme depend on the characteristic p.
- Vakil's Murphy's Law holds for this scheme.

## Abstract

We prove that the Hilbert scheme of points on a higher dimensional affine space is non-reduced and has components lying entirely in characteristic p for all primes p. In fact, we show that Vakil's Murphy's Law holds up to retraction for this scheme. Our main tool is a generalized version of the Bialynicki-Birula decomposition.