Exercise 111.39.5. Let $k$ be a field. Suppose that $X \subset \mathbf{P}^3_ k$ is a closed subscheme given by a single homogeneous equation $F \in k[X_0, X_1, X_2, X_3]$. In other words,
\[ X = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F)) \subset \mathbf{P}^3_ k \]
as explained in the course. Assume that
\[ F = X_0 G + X_1 H \]
for some homogeneous polynomials $G, H \in k[X_0, X_1, X_2, X_3]$ of positive degree. Show that if $X_0, X_1, G, H$ have no common zeros then there exists a nonconstant morphism
\[ X \longrightarrow \mathbf{P}^1_ k \]
of schemes over $\mathop{\mathrm{Spec}}(k)$ which on field points (see Exercise 111.39.1) looks like $(x_0 : x_1 : x_2 : x_3) \mapsto (x_0 : x_1)$ whenever $x_0$ or $x_1$ is not zero.
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