Detaching embedded points

TL;DR

This paper investigates the deformation and limits of space curves with embedded points, establishing conditions under which such curves are limits of simpler configurations and analyzing the structure of related Hilbert schemes.

Contribution

It demonstrates that certain space curves with embedded points are flat limits of simpler curves and points, and characterizes components of their Hilbert schemes.

Findings

Curves with embedded points of multiplicity ≤ 3 are flat limits of simpler curves.

Identifies irreducible components of Hilbert schemes for space curves with high genus.

Shows smoothness of specific Hilbert scheme components involving plane curves and points.

Abstract

We show that if $D \subset \mathbb P^N$ is obtained from a codimension two local complete intersection $C$ by adding embedded points of multiplicity $\leq 3$, then $D$ is a flat limit of $C$ and isolated points. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus, show the smoothness of the Hilbert component whose general member is a plane curve union a point in $\mathbb P^3$, and construct a Hilbert component whose general member has an embedded point.