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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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