Lemma 55.11.3. In Situation 55.9.3 with \kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i}) and g_ i the genus of C_ i the data
n, m_ i, (C_ i \cdot C_ j), [\kappa _ i : k], g_ i
is a numerical type of genus equal to the genus of C.
Proof. (In the proof of Lemma 55.11.2 we have seen that the quantities used in the statement of the lemma are well defined.) We have to verify the conditions (1) – (5) of Definition 55.3.1.
Condition (1) is immediate.
Condition (2). Symmetry of the matrix (C_ i \cdot C_ j) follows from Equation (55.9.6.1) and Lemma 55.9.6. Nonnegativity of (C_ i \cdot C_ j) for i \not= j is part (1) of Lemma 55.9.7.
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.
The genus formula proved in Lemma 55.11.2 tells us that the numerical type has the genus as stated, see Definition 55.3.4. \square
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