111.38 Morphisms from the projective line
In this section we study morphisms from $\mathbf{P}^1$ to projective schemes.
Exercise 111.38.1. 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 111.38.2. 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 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.
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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