Lemma 33.45.11 (0BEX)—The Stacks project

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Part 2: Schemes
Chapter 33: Varieties
Section 33.45: Numerical intersections
Lemma 33.45.11 (cite)

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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. $\square$

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