Lemma 33.45.11. Let $k$ be a field. Let $f : Y \to X$ be a finite dominant morphism of proper varieties over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Then

where $\deg (f)$ is as in Morphisms, Definition 29.51.8.

Lemma 33.45.11. Let $k$ be a field. Let $f : Y \to X$ be a finite dominant morphism of proper varieties over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Then

\[ \deg _{f^*\mathcal{L}}(Y) = \deg (f) \deg _\mathcal {L}(X) \]

where $\deg (f)$ is as in Morphisms, Definition 29.51.8.

**Proof.**
The statement makes sense because $f^*\mathcal{L}$ is ample by Morphisms, Lemma 29.37.7. Having said this the result is a special case of Lemma 33.45.7.
$\square$

Your email address will not be published.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.

## Comments (0)