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Tag 0DJ0

102.37. Morphisms from the projective line

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

Exercise 102.37.1. 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 102.37.2. 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 102.37.3. 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 102.37.2.

Exercise 102.37.4. 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 102.37.3 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.

    The code snippet corresponding to this tag is a part of the file exercises.tex and is located in lines 3529–3612 (see updates for more information).

    \section{Morphisms from the projective line}
    \label{section-from-P1}
    
    \noindent
    In this section we study morphisms from $\mathbf{P}^1$
    to projective schemes.
    
    \begin{exercise}
    \label{exercise-from-generic-point}
    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
    $$
    \varphi : \Spec(k(t)) \longrightarrow X
    $$
    over $k$, there exist a nonzero $f \in k[t]$ and a morphism
    $\psi : \Spec(k[t, 1/f]) \to X$ over $k$
    such that $\varphi$ is the composition
    $$
    \Spec(k(t)) \longrightarrow \Spec(k[t, 1/f]) \longrightarrow X
    $$
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-from-generic-point-Pn}
    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
    $$
    \varphi : \Spec(k(t)) \longrightarrow \mathbf{P}^n_k
    $$
    over $k$, there exists a morphism $\psi : \Spec(k[t]) \to \mathbf{P}^n_k$
    over $k$ such that $\varphi$ is the composition
    $$
    \Spec(k(t)) \longrightarrow \Spec(k[t]) \longrightarrow \mathbf{P}^n_k
    $$
    Hint: the image of $\varphi$ is in a standard open $D_+(T_i)$
    for some $i$; then show that you can ``clear denominators''.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-from-generic-point-projective}
    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
    $$
    \varphi : \Spec(k(t)) \longrightarrow X
    $$
    over $k$, there exists a morphism $\psi : \Spec(k[t]) \to X$
    over $k$ such that $\varphi$ is the composition
    $$
    \Spec(k(t)) \longrightarrow \Spec(k[t]) \longrightarrow X
    $$
    Hint: use Exercise \ref{exercise-from-generic-point-Pn}.
    \end{exercise}
    
    \begin{exercise}
    \label{exercise-from-generic-point-P1-to-projective}
    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
    $$
    \varphi : \Spec(K) \longrightarrow X
    $$
    over $k$, there exists a morphism $\psi : \mathbf{P}^1_k \to X$
    over $k$ such that $\varphi$ is the composition
    $$
    \Spec(k(t)) \longrightarrow \mathbf{P}^1_k \longrightarrow X
    $$
    Hint: use Exercise \ref{exercise-from-generic-point-projective}
    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.
    \end{exercise}

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