Theorem NF: a 2×3 configuration with ratio of degree 2 in a five-dimensional W ⊇ span(1, u, u⁻¹) makes W complete, two-orbit or cyclic

Let $u \in \mathbb{C}$ be transcendental, that is, $u \notin \overline{\mathbb{Q}}$ (the algebraic numbers); spans and linear independence are over $\overline{\mathbb{Q}}$. Let $W \subseteq \mathbb{C}$ be a $\overline{\mathbb{Q}}$-vector subspace of dimension $5$ with $1, u, u^{-1} \in W$, that is, $H_0 \subseteq W$ for $H_0 = \mathrm{span}(1, u, u^{-1})$. Let $x_0, x_1 \in \mathbb{C}$ be linearly independent and $y_0, y_1, y_2 \in \mathbb{C}$ linearly independent, with $x_iy_j \in W$ for all $i, j$ (a $2 \times 3$ configuration in $W$). Let $p, q \in \overline{\mathbb{Q}}[X]$ be coprime with $\max(\deg p, \deg q) = 2$, and suppose $x_1q(u) = x_0p(u)$, so that the ratio $x_1/x_0$ is $p(u)/q(u)$. Then one of the following holds:

• (D1) there are linearly independent $t_0, t_1, t_2 \in \mathbb{C}$ with $t_j \in W$ and $ut_j \in W$ for every $j$;
• (D2a, two-orbit) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), P'(u)S_2(u)/(uQ'(u)))$ for some $P', Q', S_1, S_2 \in \overline{\mathbb{Q}}[X]$ with $P', Q'$ coprime, $\max(\deg P', \deg Q') = 2$ and $\deg S_1, \deg S_2 \leq 2$;
• (D2b, cyclic) $W = \mathrm{span}(1, u, u^{-1}, Q'(u)S_1(u)/(uP'(u)), Q'(u)(Q'(u)S_1(u) + P'(u)S_2(u))/(uP'(u)^2))$ for some such $P', Q', S_1, S_2$.

$W$ is not assumed to consist of rational functions of $u$; in cases (D2a) and (D2b) the conclusion shows that it does.

Proof. Put $h = p(u)/q(u)$. As $x$ is independent, $x_0 \neq 0$; and $q(u) \neq 0$, for otherwise $x_1q(u) = x_0p(u)$ gives $p(u) = 0$ as well, against a Bézout identity $ap + bq = 1$. So $x_1 = hx_0$, and $y' = x_0y$ is an independent triple (multiplication by $x_0 \neq 0$ is injective) with $y'_j \in W$ and $hy'_j = x_1y_j \in W$. DiazModulus.degree_two_configuration_multiplier_complement gives $\lambda \in \{0, 1, 2\}$ and $g_1, g_2$ with $W = \mathrm{span}(1, u, u^{-1}, g_1, g_2)$ and $kg_j \in W$ for $k = q(u)/(p(u) - \lambda q(u))$. The pencil $P = q$, $Q = p - \lambda q$ is coprime of maximal degree $2$ by DiazModulus.coprime_pencil_mobius (determinant $-1$), and $k = P(u)/Q(u)$. Then DiazModulus.quadratic_complement_eigen_forms, applied to $(g_1, g_2)$ with $S = W$, gives the scalar, two-orbit or cyclic shape, and DiazModulus.quadratic_eigen_forms_canonical turns it into (D1), (D2a) or (D2b).

Novelty. Not found in the sources read; the statement is Theorem 2 of the project's own research notes, proved there with vector bundles on $\mathbb{P}^1$ (D2a and D2b are its normal forms after a change of coordinate). The step from a configuration spanning four dimensions to a progression is Lemma 4 of Bachoc, Serra and Zémor (2017). No source on scrolls or on vector bundles on $\mathbb{P}^1$ was among those read.