Consider the family of all conics in $\mathbb{P}^{3}_{k}$, with $k$ an algebraically closed field. Such curves all have degree $2$ and genus $0$, and they can be uniquely defined to be all curves in $\mathbb{P}^{3}_{k}$ with Hilbert polynomial $p(t)=2t+1$.
My goal is to study the corresponding Chow variety $C^{2t+1}_3$. In particular, for any double line $L$, I would like to compute the corresponding element in the Chow variety, and show that the scheme structure on such duble line is not taken into account, when constructing the Chow variety.