It is well known that lines of the projective space $\mathbb P^3$ — or
equivalently, 2-dimensional subspaces of a 4-dimensional vector space — are
parametrized by a projective variety of dimension four, the Grassmannian
$G(2,4)$. And Plücker coordinates realize this variety as a quadric of
$\mathbb P^5$. What about the parametrization curves of degree two in $\mathbb
P^3$? Celebrated predecessors — Cayley, van der Waerden, Green, Morrison,
Gel’fand, Kapranov, Zelevinsky, *etc.* — have shown how to realize this
parametrization as a 8-dimensional subvariety of $\mathbb P^{19}$: the Chow
variety of quadratic space curves.

In “Computing the Chow variety of quadratic space curves”, we performed the computations all the way through to give the explicit equations. For example, we obtain that the two components of the Chow variety of quadratic space curves have degree 140 and 92.