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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 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.)

    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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