Irrational components of the Hilbert scheme of points

TL;DR

This paper constructs irrational irreducible components of the Hilbert scheme of points in high-dimensional affine space, extending known results from lower dimensions and solving a specific open problem in the field.

Contribution

It introduces a method to produce irrational components of Hilbert schemes of points in dimensions at least 12, linking curve components in P^3 to higher-dimensional cases.

Findings

Irrational components exist in Hilbert schemes of points for n ≥ 12.

Methods relate curve components in P^3 to higher-dimensional schemes.

The work resolves an open problem posed by Jelisiejew in 2023.

Abstract

We construct irrational irreducible components of the Hilbert scheme of points of affine n-dimensional space, for n at least 12. We start with irrational components of the Hilbert scheme of curves in P^3 and use methods developed by Jelisiejew to relate these to irreducible components of the Hilbert schemes of points of A^n. The result solves Problem XX of [J. Jelisiejew, Open problems in deformations of Artinian algebras, Hilbert schemes and around, arXiv:2307.08777, 2023].