My question concerns the relationship between chow varieties and hilbert schemes in the case of conics in $\mathbb{P}^{3}_{k}$. More precisely, consider the Hilbert scheme $\mathsf{hilb}^{2t+1}_{3}$, and the Chow variety $C^{2t+1}_{3}$.
Let $f:\mathsf{hilb}^{2t+1}_{3} \longrightarrow C^{2t+1}_{3}$ be the rational map that maps every reduced element $X$ to the corresponding Chow variety $V_{X}$. I would like to show that this map is not an isomorphism on the reduced elements $X$ that correspond to double lines.
My overall goal is to give a concrete example of family of curves, for which the correspinding Chow variety and Hilbert scheme behave differently.