Bertini for normality: a general hyperplane section of a normal variety is normal Let $X$ be a normal (complex) projective variety. There is an old paper by Seidenberg which shows that a general hyperplane section of $X$ is normal.
Does anyone know a modern proof (or reference) for this result?
 A: This is a cross-site duplicate of this MO question. This is a community-wiki post recording the accepted answer to that question by Takumi Murayama. The section below the horizontal separator is a copy-paste of that post.

I needed to know the answer to this myself, so here is a good reference:

Hubert Flenner, Liam O’Carroll, and Wolfgang Vogel, Joins and
intersections, Springer Monographs in Mathematics, Springer-Verlag,
Berlin, 1999. MR 1724388 DOI 10.1007/978-3-662-03817-8

The relevant results are the following:
Theorem 3.4.10. Let $X$ be a variety over an infinite field $K$, and let $D$ be a Cartier divisor on $X$. Assume that $\Gamma \subseteq \lvert D \rvert$ is a finite-dimensional linear system that is not composed with a pencil and satisfies $\operatorname{codim} \operatorname{Bs}(\Gamma) \ge 2$. Then, a generic member of $\Gamma$ is irreducible.
Corollary 3.4.14. Let $X \subseteq \mathbf{P}^n_K$ be a projective scheme over an infinite field $K$ which is regular (resp. normal, reduced, satisfies $R_k$). Then, for a generic hyperplane $H \subseteq \mathbf{P}^n_K$, the intersection $X \cap H$ has the same property.
