contraction and stabilization from [@Knud]. We describe a related construction, the contraction to the $i$-component, of which the contraction to the $v$-component mentioned above is a special case.

**Section 2.** We recall the construction of $B_V$ from [@P-Sch] and gather additional results from *loc. cit.* which will be of use to us. In particular, we describe the open subscheme $U_{{\cal F}}\subset B_V$ associated to a flag ${{\cal F}}$ of $V$.

**Section 3.** The concept of $V$-ferns is formally introduced. We then demonstrate several of their properties. First, we discuss $V$-ferns over a field, and show that there is a natural way to associate a flag of $V$ to each such fern. As a result, for a $V$-fern over an arbitrary ${{\mathbb{F}}}_q$-scheme $S$, we can associate a flag ${{\cal F}}_s$ of $V$ to each $s\in S$. We then discuss $V$-ferns for $\dim V=1$ and $2$. In the first case, we show that there is exactly one $V$-fern over $S$ up to (unique) isomorphism. This provides the base case for the induction in our proof of the Main Theorem.

**Section 4.** We describe the contraction, grafting and line bundle constructions associated to $V$-ferns in detail.

**Section 5.** Let ${{\cal F}}$ be a 