the $U_{{\cal F}}$ form an open covering of $B_V$.

In order to endow ${{\cal C}}_V$ with the structure of a $V$-fern, we separately consider a scheme ${{\cal C}}_{{\cal F}}$, which is a closed subscheme of $U_{{\cal F}}\times P^\Sigma$ defined by explicit polynomial equations. We endow ${{\cal C}}_{{\cal F}}$ with the structure of a $V$-fern and then show that ${{\cal C}}_{{\cal F}}= \pi^{-1}(U_{{\cal F}})$. This allows us to glue the $V$-fern structures on the various ${{\cal C}}_{{\cal F}}$ to obtain one on ${{\cal C}}_V.$

### Step 2: Reduction to the case of ${{\cal F}}$-ferns {#step-2-reduction-to-the-case-of-cal-f-ferns .unnumbered}

We show that for a $V$-fern $(C,\lambda,\phi)$ over a scheme $S$ one can associate to each $s\in S$ a flag ${{\cal F}}_s$ coming from the fiber $C_s$. For a fixed flag ${{\cal F}}$, we define an *${{\cal F}}$-fern* to be a $V$-fern for which ${{\cal F}}_s\subset{{\cal F}}$. The locus $S_{{\cal F}}:=\{s\in S\mid {{\cal F}}_s\subset {{\cal F}}\}$ is open in $S$ and the $S_{{\cal F}}$ cover $S$ as ${{\cal F}}$ varies over all (complete) flags of $V$. We reduce the proof of the Main Theorem to showing that $U_{{\cal F}}$ represents $\Fern_{{\cal F}}$, where ${{\cal F}}$ is a complete flag, with the universal ${{\cal F}}$-fern is 