# The space of twisted cubics

## TL;DR

This paper studies the moduli space of twisted cubics in projective space, showing its compactification and relation to the Hilbert scheme, providing a detailed geometric description.

## Contribution

It introduces a Cohen-Macaulay compactification of the space of twisted cubics and establishes its isomorphism with a component of the Hilbert scheme in projective 3-space.

## Key findings

- The moduli scheme of CM-curves in P^3 is isomorphic to the twisted cubic component of the Hilbert scheme.
- The paper describes the compactification of twisted cubics in higher-dimensional projective spaces.
- Provides a detailed geometric structure of the moduli space of twisted cubics.

## Abstract

We consider the Cohen-Macaulay compactification of the space of twisted cubics in projective n-space. This compactification is the fine moduli scheme representing the functor of CM-curves with Hilbert polynomial 3t+1. We show that the moduli scheme of CM-curves in projective 3-space is isomorphic to the twisted cubic component of the Hilbert scheme. We also describe the compactification for twisted cubics in n-space.