Hilbert scheme of twisted cubics as simple wall-crossing

TL;DR

This paper investigates the Hilbert scheme of twisted cubics in projective space through Bridgeland stability and wall-crossing, revealing geometric structures and singularities, and reestablishing classical results.

Contribution

It applies Bridgeland stability and wall-crossing techniques to analyze the Hilbert scheme of twisted cubics, providing new geometric insights and reproving classical theorems.

Findings

Description of the geometric structure of the Hilbert scheme

Identification of singularities within the scheme

Reproof of classical results by Piene and Schlessinger

Abstract

We study the Hilbert scheme of twisted cubics in the three-dimensional projective space by using Bridgeland stability conditions. We use wall-crossing techniques to describe its geometric structure and singularities, which reproves the classical result of Piene and Schlessinger.