Lemma 55.11.2. In Situation 55.9.3 we have
where \kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i}), g_ i is the genus of C_ i, and g_ C is the genus of C.
Lemma 55.11.2. In Situation 55.9.3 we have
where \kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i}), g_ i is the genus of C_ i, and g_ C is the genus of C.
Proof. Our basic tool will be Derived Categories of Schemes, Lemma 36.32.2 which shows that
Choose a sequence of effective Cartier divisors
such that D_{j + 1} = D_ j + C_{i_ j} for each j. (It is clear that we can choose such a sequence by decreasing one nonzero multiplicity of D_{j + 1} one step at a time.) Applying Lemma 55.11.1 starting with \chi (\mathcal{O}_{D_0}) = 0 we get
Perhaps the last equality deserves some explanation. Namely, since \sum _ j C_{i_ j} = \sum m_ i C_ i we have (\sum _ j C_{i_ j} \cdot \sum _ j C_{i_ j}) = 0 by Lemma 55.9.7. Thus we see that
by splitting this product into “nondiagonal” and “diagonal” terms. Note that \kappa _ i is a field finite over k by Varieties, Lemma 33.26.2. Hence the genus of C_ i is defined and we have \chi (C_ i, \mathcal{O}_{C_ i}) = [\kappa _ i : k](1 - g_ i). Putting everything together and rearranging terms we get
which is what the lemma says too. \square
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