The Stacks project

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.