\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

The Stacks project

103.37 Morphisms from the projective line

In this section we study morphisms from $\mathbf{P}^1$ to projective schemes.

Exercise 103.37.1. 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 finite type scheme over $k$. Show that for any morphism

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

over $k$, there exist a nonzero $f \in k[t]$ and a morphism $\psi : \mathop{\mathrm{Spec}}(k[t, 1/f]) \to X$ over $k$ such that $\varphi $ is the composition

\[ \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathop{\mathrm{Spec}}(k[t, 1/f]) \longrightarrow X \]

Exercise 103.37.2. Let $k$ be a field. Let $k[t] \subset k(t)$ be the inclusion of the polynomial ring into its fraction field. Show that for any morphism

\[ \varphi : \mathop{\mathrm{Spec}}(k(t)) \longrightarrow \mathbf{P}^ n_ k \]

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

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

Hint: the image of $\varphi $ is in a standard open $D_+(T_ i)$ for some $i$; then show that you can “clear denominators”.

Exercise 103.37.3. 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 103.37.2.

Exercise 103.37.4. 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 103.37.3 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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