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

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$^{4m+1}({\bold P}^4)$.