Exercise 111.38.3 (0DJ3)—The Stacks project

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Part 9: Miscellany
Chapter 111: Exercises
Section 111.38: Morphisms from the projective line
Exercise 111.38.3 (cite)

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Exercise 111.38.3. Let $k$ be a field. Let $k[t] \subset k(t)$ be the inclusion of the polynomial ring into its fraction field. Let $X$ be a projective scheme over $k$ . Show that for any morphism

$\varphi : \mathop{\mathrm{Spec}}(k(t)) \longrightarrow X$

over $k$ , there exists a morphism $\psi : \mathop{\mathrm{Spec}}(k[t]) \to X$ over $k$ such that $\varphi$ is the composition

$\mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathop{\mathrm{Spec}}(k[t]) \longrightarrow X$

Hint: use Exercise 111.38.2.

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