Lemma 33.45.12 (0BEY)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.45: Numerical intersections
Lemma 33.45.12 (cite)

previous lemma
next proposition

numberstags

history
statistics

Lemma 33.45.12. Let $k$ be a field. Let $X$ be a proper scheme over $k$ . Let $Z \subset X$ be a closed subscheme of dimension $\leq 1$ . Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$ -module. Then

$(\mathcal{L} \cdot Z) = \deg (\mathcal{L}|_ Z)$

where $\deg (\mathcal{L}|_ Z)$ is as in Definition 33.44.1. If $\mathcal{L}$ is ample, then $\deg _\mathcal {L}(Z) = \deg (\mathcal{L}|_ Z)$ .

Proof. This follows from the fact that the function $n \mapsto \chi (Z, \mathcal{L}|_ Z^{\otimes n})$ has degree $1$ and hence the leading coefficient is the difference of consecutive values. $\square$

Comments (0)

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).

Comments (0)

Post a comment