## Tag `02AA`

Chapter 102: Exercises > Section 102.38: Morphisms from surfaces to curves

Exercise 102.38.4. Show there does not exist a nonconstant morphism of schemes $\mathbf{P}^2_{\mathbf{C}} \to \mathbf{P}^1_{\mathbf{C}}$ over $\mathop{\mathrm{Spec}}(\mathbf{C})$. Here a

constant morphismis one whose image is a single point. (Hint: If the morphism is not constant consider the fibres over $0$ and $\infty$ and argue that they have to meet to get a contradiction.)

The code snippet corresponding to this tag is a part of the file `exercises.tex` and is located in lines 3654–3662 (see updates for more information).

```
\begin{exercise}
\label{exercise-no-nonconstant-morphism-proj}
Show there does not exist a nonconstant morphism of schemes
$\mathbf{P}^2_{\mathbf{C}} \to \mathbf{P}^1_{\mathbf{C}}$
over $\Spec(\mathbf{C})$. Here a {\it constant morphism} is
one whose image is a single point.
(Hint: If the morphism is not constant consider the fibres over
$0$ and $\infty$ and argue that they have to meet to get a contradiction.)
\end{exercise}
```

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