The Stacks project

Exercise 111.38.4. Let $k$ be a field. Let $X$ be a projective scheme over $k$. Let $K$ be the function field of $\mathbf{P}^1_ k$ (see hint below). Show that for any morphism

\[ \varphi : \mathop{\mathrm{Spec}}(K) \longrightarrow X \]

over $k$, there exists a morphism $\psi : \mathbf{P}^1_ k \to X$ over $k$ such that $\varphi $ is the composition

\[ \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathbf{P}^1_ k \longrightarrow X \]

Hint: use Exercise 111.38.3 for each of the two pieces of the affine open covering $\mathbf{P}^1_ k = D_+(T_0) \cup D_+(T_1)$, use that $D_+(T_0)$ is the spectrum of a polynomial ring and that $K$ is the fraction field of this polynomial ring.

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