Generically nonreduced components of Hilbert schemes on fourfolds

TL;DR

This paper demonstrates the existence of nonreduced components in the Hilbert scheme of points on fourfolds, providing new insights into the scheme's structure and answering an open problem.

Contribution

It introduces a method to identify generically nonreduced components of Hilbert schemes on fourfolds, advancing understanding of their geometric properties.

Findings

Existence of nonreduced components for at least 21 points on fourfolds

Method similar to previous approaches for analyzing Hilbert schemes

Answers an open problem posed in prior research

Abstract

We exhibit generically nonreduced components of the Hilbert scheme of at least $21$ points on a smooth variety of dimension at least four. The result was announced in~[Jelisiejew__open_problems] and answers a question~[Problem~3.8, AIMPL]. The method is similar to the one of~[\S6, Jelisiejew-\v{S}ivic].