is a numerical type of genus equal to the genus of $C$.
Condition (1) is immediate.
Condition (3) is part (3) of Lemma 55.9.7.
Condition (4) is part (2) of Lemma 55.9.7.
Condition (5) follows from the fact that $(C_ i \cdot C_ j)$ is the degree of an invertible module on $C_ i$ which is divisible by $[\kappa _ i : k]$, see Varieties, Lemma 33.44.10.
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