given by ${{\cal C}}_{{\cal F}}$.

### Step 3: Morphism to $U_{{\cal F}}$ representing an ${{\cal F}}$-fern {#step-3-morphism-to-u_cal-f-representing-an-cal-f-fern .unnumbered}

Let ${{\cal F}}$ be a complete flag and let $(C,\lambda,\phi)$ be an ${{\cal F}}$-fern over a scheme $S$. Using one of the constructions described above, we associate a pair $(L,\lambda)$ to $C$ consisting of a line bundle $L$ over $S$ and an ${{\mathbb{F}}}_q$-linear map $\lambda\colon V\to L(S)$ such that the image of $V$ generates the sheaf of sections ${{\cal L}}$ of $L$. Using the well-known description of the functor of points of projective space ([@Ha §II.7]), the isomorphism class of the pair $(L,\lambda)$ corresponds to a unique morphism $S\to P_V$. By repeatedly contracting and applying the same construction, we obtain morphisms $S\to P_{V'}$ for each $0\neq V'\subset V$, and hence a morphism $$f_C\colon S\to\prod_{0\neq V'\subset V}P_{V'}.$$ The scheme $U_{{\cal F}}$ is a locally closed subscheme of $\prod_{0\neq V'\subset V}P_{V'}$, and we show that $f_C$ factors through $U_{{\cal F}}$.

### Step 4: Key lemma for induction {#step-4-key-lemma-for-induction .unnumbered}

Write ${{\cal F}}=\{V_0,\ldots V_n\}$ and ${{\cal F}}'=\{V_0,\ldots V_{n-1}=:V'\}$. We prove the following lemma:

Let $(C,\lambda,\phi)$ and $(D,\mu,\psi)$ be ${{\cal F}}$-ferns such that the $V'$-contractions $C'$ and $D'$ are isomorphic ${{\cal F}}'$-ferns. Suppose further that $C^\infty\cong 