Remark 37.32.4. Let us sketch a “geometric” proof of a special case of Lemma 37.32.3. Namely, say $k$ is an algebraically closed field and $X \subset \mathbf{P}^ n_ k$ is smooth and irreducible of dimension $\geq 2$. Then we claim there is a hyperplane $H \subset \mathbf{P}^ n_ k$ such that $X \cap H$ is smooth and irreducible. Namely, by Varieties, Lemma 33.47.3 for a general $v \in V = kT_0 \oplus \ldots \oplus kT_ n$ the corresponding hyperplane section $X \cap H_ v$ is smooth. On the other hand, by Enriques-Severi-Zariski the scheme $X \cap H_ v$ is connected, see Varieties, Lemma 33.48.3. Hence $X \cap H_ v$ is smooth and irreducible.

## 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Your email address will not be published.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)