The Stacks project

Lemma 55.9.7. In Situation 55.9.3 the symmetric bilinear form (55.9.6.1) has the following properties

  1. $(C_ i \cdot C_ j) \geq 0$ if $i \not= j$ with equality if and only if $C_ i \cap C_ j = \emptyset $,

  2. $(\sum m_ i C_ i \cdot C_ j) = 0$,

  3. there is no nonempty proper subset $I \subset \{ 1, \ldots , n\} $ such that $(C_ i \cdot C_ j) = 0$ for $i \in I$, $j \not\in I$.

  4. $(\sum a_ i C_ i \cdot \sum a_ i C_ i) \leq 0$ with equality if and only if there exists a $q \in \mathbf{Q}$ such that $a_ i = qm_ i$ for $i = 1, \ldots , n$,

Proof. In the proof of Lemma 55.9.6 we saw that $(C_ i \cdot C_ j) = \deg (C_ i \cap C_ j)$ if $i \not= j$. This is $\geq 0$ and $> 0 $ if and only if $C_ i \cap C_ j \not= \emptyset $. This proves (1).

Proof of (2). This is true because by Lemma 55.9.1 the invertible sheaf associated to $\sum m_ i C_ i$ is trivial and the trivial sheaf has degree zero.

Proof of (3). This is expressing the fact that $X_ k$ is connected (Lemma 55.9.4) via the description of the intersection products given in the proof of (1).

Part (4) follows from (1), (2), and (3) by Lemma 55.2.3. $\square$

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