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

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

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

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

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

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