The Stacks project

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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DJ0. Beware of the difference between the letter 'O' and the digit '0'.