Lemma 55.9.10. In Situation 55.9.3 let d = \gcd (m_1, \ldots , m_ n). Let D = \sum (m_ i/d) C_ i as an effective Cartier divisor on X. Then
where g_ C is the genus of C, g_ D is the genus of D, and \kappa = H^0(D, \mathcal{O}_ D).
[Lemma 2.6, Artin-Winters]
Lemma 55.9.10. In Situation 55.9.3 let d = \gcd (m_1, \ldots , m_ n). Let D = \sum (m_ i/d) C_ i as an effective Cartier divisor on X. Then
where g_ C is the genus of C, g_ D is the genus of D, and \kappa = H^0(D, \mathcal{O}_ D).
Proof. By Lemma 55.9.9 we see that \kappa is a field and a finite extension of k. Since also H^0(C, \mathcal{O}_ C) = K we see that the genus of C and D are defined (see Algebraic Curves, Definition 53.8.1) and we have g_ C = \dim _ K H^1(C, \mathcal{O}_ C) and g_ D = \dim _\kappa H^1(D, \mathcal{O}_ D). By Derived Categories of Schemes, Lemma 36.32.2 we have
We claim that
This will prove the lemma because
Observe that X_ k = dD as an effective Cartier divisor. To prove the claim we prove by induction on 1 \leq r \leq d that \chi (rD, \mathcal{O}_{rD}) = r \chi (D, \mathcal{O}_ D). The base case r = 1 is trivial. If 1 \leq r < d, then we consider the short exact sequence
of Divisors, Lemma 31.14.3. By additivity of Euler characteristics (Varieties, Lemma 33.33.2) it suffices to prove that \chi (D, \mathcal{O}_ X(rD)|_ D) = \chi (D, \mathcal{O}_ D). This is true because \mathcal{O}_ X(rD)|_ D is a torsion element of \mathop{\mathrm{Pic}}\nolimits (D) (Lemma 55.9.8) and because the degree of a line bundle is additive (Varieties, Lemma 33.44.7) hence zero for torsion invertible sheaves. \square
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