The Stacks project

Lemma 55.11.1. In Situation 55.9.3 suppose we have an effective Cartier divisors $D, D' \subset X$ such that $D' = D + C_ i$ for some $i \in \{ 1, \ldots , n\} $ and $D' \subset X_ k$. Then

\[ \chi (X_ k, \mathcal{O}_{D'}) - \chi (X_ k, \mathcal{O}_ D) = \chi (X_ k, \mathcal{O}_ X(-D)|_{C_ i}) = -(D \cdot C_ i) + \chi (C_ i, \mathcal{O}_{C_ i}) \]

Proof. The second equality follows from the definition of the bilinear form $(\ \cdot \ )$ in (55.9.6.1) and Lemma 55.9.6. To see the first equality we distinguish two cases. Namely, if $C_ i \not\subset D$, then $D'$ is the scheme theoretic union of $D$ and $C_ i$ (by Divisors, Lemma 31.13.10) and we get a short exact sequence

\[ 0 \to \mathcal{O}_{D'} \to \mathcal{O}_ D \times \mathcal{O}_{C_ i} \to \mathcal{O}_{D \cap C_ i} \to 0 \]

by Morphisms, Lemma 29.4.6. Since we also have an exact sequence

\[ 0 \to \mathcal{O}_ X(-D)|_{C_ i} \to \mathcal{O}_{C_ i} \to \mathcal{O}_{D \cap C_ i} \to 0 \]

(Divisors, Remark 31.14.11) we conclude that the claim holds by additivity of euler characteristics (Varieties, Lemma 33.33.2). On the other hand, if $C_ i \subset D$ then we get an exact sequence

\[ 0 \to \mathcal{O}_ X(-D)|_{C_ i} \to \mathcal{O}_{D'} \to \mathcal{O}_ D \to 0 \]

by Divisors, Lemma 31.14.3 and we immediately see the lemma holds. $\square$

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