Arithmetically Cohen--Macaulay curves in P^4 of degree 4 and genus 0

TL;DR

This paper investigates ACM curves of degree 4 and genus 0 in projective 4-space, showing their irreducibility in the Hilbert scheme and analyzing the structure and singularities of their moduli space.

Contribution

It proves the irreducibility of the ACM curves of degree 4 and genus 0 in P^4 and describes the structure of all such curves within their Hilbert scheme component.

Findings

ACM curves of degree 4, genus 0 form an irreducible subset of the Hilbert scheme.

The singular locus of the Hilbert scheme component has dimension greater than 6.

Complete description of all ACM curves of degree 4 and genus 0 in P^4.

Abstract

We show that the arithmetically Cohen--Macaulay (ACM) curves of degree 4 and genus 0 in ${\bold P}^4$ form an irreducible subset of the Hilbert scheme. Using this, we show that the singular locus of the corresponding component of the Hilbert scheme has dimension greater than 6. Moreover, we describe the structures of all ACM curves of Hilb$^{4m+1}({\bold P}^4)$.