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 33.44.7. 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 33.44.12.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)