The Cohen-Macaulay space of twisted cubics

In this work, we describe the Cohen-Macaulay space CM of twisted cubics parameterizing curves $C$ together with a finite map $i: C \to \mathbb{P}^3$ that is generically a closed immersion and such that $C$ has Hilbert polynomial $p(t)=3t+1$ with respect to $i$. We show that CM is irreducible, smooth and birational to one component of the Hilbert scheme of twisted cubics.