in the Main Theorem by $$\begin{array}{cccl}
\Fern_V\colon&\SchFqOp&\longto&\Set\\
&S&\mapsto&\left\{\substack{\mbox{Isomorphism classes} \\
    \mbox{of $V$-ferns over $S$}}\right\}.
\end{array}$$ Roughly speaking, our strategy for proving the theorem is to begin by defining a universal $V$-fern $({{\cal C}}_V,\lambda_V,\phi_V)$ over $B_V$. For an arbitrary ${{\mathbb{F}}}_q$-scheme $S$ and a $V$-fern $(C,\lambda,\phi)$ over $S$, we then show that there exists a unique morphism $f_C\colon S\to B_V$ such that $C$ is isomorphic to the pullback $f_C^*{{\cal C}}_V$. We now give a more detailed outline of the proof.

### Step 1: Construct the universal family {#step-1-construct-the-universal-family .unnumbered}

Let $\Sigma:=V\times (V\setminus\{0\})$ and let $P^\Sigma:=\prod_{(v,w)\in \Sigma}{{\mathbb{P}}}^1.$ We define the universal $V$-fern $\pi\colon {{\cal C}}_V\to B_V$ as the scheme theoretic closure of the image of a certain morphism from $\OV\times{{\mathbb{A}}}^1$ to $B_V\times P^\Sigma$.

Recall that a *flag* ${{\cal F}}$ of $V$ is a set $\{V_0,\ldots,V_m\}$ of subspaces of $V$ with $m\in {{\mathbb{Z}}}_{>0}$ such that $V_0=\{0\}$ and $V_m=V$ and $V_i\subsetneq V_j$ for $i<j$. We say that ${{\cal F}}$ is *complete* if $m=\dim V$. For each flag ${{\cal F}}$ of $V$, there is an open subscheme $U_{{\cal F}}$ of $B_V$ which represents a certain subfunctor of the functor in [@P-Sch] which is represented by $B_V$. As ${{\cal F}}$ varies over all (complete) flags of $V$, 