I'm not so strong good on the scheme-theoretic language, so let me embed $F_n$ as the affine variety $latex \text{res}{n,n}(X^n \text{res}_{n,n}(X^n + ..., b{n-1} b_{n-1} X^{n-1} + ...) y = 1$ in one dimension up. Then a morphism $\text{Spec } k[a_0, k[a_0, ... a_{n-1}, b_0, ... b_{n-1}, y]$y]/(\text{stuff}) \to R$ is precisely (assuming that Cazanava means either $k = \mathbb{Z}$ or $R$ a $k$-algebra) a choice, for each variable $a_i, b_i, y$, of an element of $R$ subject to the condition that the resultant times $y$ is equal to $1$, i.e. the resultant is invertible in $R$.

I'm not so strong on the scheme-theoretic language, so let me embed $F_n$ as the affine variety $latex \text{res}{n,n}(X^n + ..., b{n-1} X^{n-1} + ...) y = 1$ in $\text{Spec } k[a_0, ... a_{n-1}, b_0, ... b_{n-1}, y]$.