The Stacks project

Lemma 54.7.4. In Situation 54.7.1 assume $X$ is normal. Let $Z \subset X$ be a nonempty effective Cartier divisor such that $Z \subset X_ s$ set theoretically. Then the conormal sheaf of $Z$ is not trivial. More precisely, there exists an $i$ such that $C_ i \subset Z$ and $\deg (\mathcal{C}_{Z/X}|_{C_ i}) > 0$.

Proof. We will use the observations made following Situation 54.7.1 without further mention. Let $f$ be a function as in Lemma 54.7.3. Let $\xi _ i \in C_ i$ be the generic point. Let $\mathcal{O}_ i$ be the local ring of $X$ at $\xi _ i$. Then $\mathcal{O}_ i$ is a discrete valuation ring. Let $e_ i$ be the valuation of $f$ in $\mathcal{O}_ i$, so $e_ i > 0$. Let $h_ i \in \mathcal{O}_ i$ be a local equation for $Z$ and let $d_ i$ be its valuation. Then $d_ i \geq 0$. Choose and fix $i$ with $d_ i/e_ i$ maximal (then $d_ i > 0$ as $Z$ is not empty). Replace $f$ by $f^{d_ i}$ and $Z$ by $e_ iZ$. This is permissible, by the relation $\mathcal{O}_ X(e_ i Z) = \mathcal{O}_ X(Z)^{\otimes e_ i}$, the relation between the conormal sheaf and $\mathcal{O}_ X(Z)$ (see Divisors, Lemmas 31.14.4 and 31.14.2, and since the degree gets multiplied by $e_ i$, see Varieties, Lemma 33.43.7. Let $\mathcal{I}$ be the ideal sheaf of $Z$ so that $\mathcal{C}_{Z/X} = \mathcal{I}|_ Z$. Consider the image $\overline{f}$ of $f$ in $\Gamma (Z, \mathcal{O}_ Z)$. By our choices above we see that $\overline{f}$ vanishes in the generic points of irreducible components of $Z$ (these are all generic points of $C_ j$ as $Z$ is contained in the special fibre). On the other hand, $Z$ is $(S_1)$ by Divisors, Lemma 31.15.6. Thus the scheme $Z$ has no embedded associated points and we conclude that $\overline{f} = 0$ (Divisors, Lemmas 31.4.3 and 31.5.6). Hence $f$ is a global section of $\mathcal{I}$ which generates $\mathcal{I}_{\xi _ i}$ by construction. Thus the image $s_ i$ of $f$ in $\Gamma (C_ i, \mathcal{I}|_{C_ i})$ is nonzero. However, our choice of $f$ guarantees that $s_ i$ has a zero at $x_ i$. Hence the degree of $\mathcal{I}|_{C_ i}$ is $> 0$ by Varieties, Lemma 33.43.12. $\square$


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