The Stacks project

Lemma 33.48.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{L}$ be an ample invertible $\mathcal{O}_ X$-module. Let $s \in \Gamma (X, \mathcal{L})$. Assume

  1. $s$ is a regular section (Divisors, Definition 31.14.6),

  2. for every closed point $x \in X$ we have $\text{depth}(\mathcal{O}_{X, x}) \geq 2$, and

  3. $X$ is connected.

Then the zero scheme $Z(s)$ of $s$ is connected.

Proof. Since $s$ is a regular section, so is $s^ n \in \Gamma (X, \mathcal{L}^{\otimes n})$ for all $n > 1$. Moreover, the inclusion morphism $Z(s) \to Z(s^ n)$ is a bijection on underlying topological spaces. Hence if $Z(s)$ is disconnected, so is $Z(s^ n)$. Now consider the canonical short exact sequence

\[ 0 \to \mathcal{L}^{\otimes -n} \xrightarrow {s^ n} \mathcal{O}_ X \to \mathcal{O}_{Z(s^ n)} \to 0 \]

Consider the $k$-algebra $R_ n = \Gamma (X, \mathcal{O}_{Z(s^ n)})$. If $Z(s)$ is disconnected, i.e., $Z(s^ n)$ is disconnected, then either $R_ n$ is zero in case $Z(s^ n) = \emptyset $ or $R_ n$ contains a nontrivial idempotent in case $Z(s^ n) = U \amalg V$ with $U, V \subset Z(s^ n)$ open and nonempty (the reader may wish to consult Lemma 33.9.3). Thus the map $\Gamma (X, \mathcal{O}_ X) \to R_ n$ cannot be an isomorphism. It follows that either $H^0(X, \mathcal{L}^{\otimes -n})$ or $H^1(X, \mathcal{L}^{\otimes -n})$ is nonzero for infinitely many positive $n$. This contradicts Lemma 33.48.1 or 33.48.2 and the proof is complete. $\square$

Comments (3)

Comment #5450 by Laurent Moret-Bailly on

Typos in proof: Line 2: "if is disconnected..." Exact sequence: should be . Line -3: "either or ".

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0FD9. Beware of the difference between the letter 'O' and the digit '0'.