Singularities of the Hilbert scheme of non-reduced curves

TL;DR

This paper investigates the non-reduced structure of the Hilbert scheme of certain curves in projective 3-space, revealing new types of infinitesimal deformations and introducing the concept of curve extension.

Contribution

It demonstrates the existence of non-reduced curves with special deformations and introduces the notion of curve extension to analyze these deformations.

Findings

Existence of non-reduced curves with non-deforming infinitesimal deformations

Introduction of the concept of extension of curves

Examples illustrating curve extensions

Abstract

In this article, we study the Hilbert scheme of generically non-reduced curves in $\mathbb{P}^3$. We prove the existence of generically non-reduced curves in $\mathbb{P}^3$ for which there exist infinitesimal deformations of the curve that do not induce deformations of the associated reduced scheme. We show that such infinitesimal deformations contribute to the non-reducedness of the corresponding Hilbert scheme. We introduce the notion of extension of curves and prove that such infinitesimal deformations (such that the associated reduced scheme does not deform) are inherited by the extended curve. Finally, we give examples of extension of curves.