Let $k$ be an algebraically closed field (of arbitrary characteristic), $X$ a projective $k$-variety (= integral projective $k$-scheme) and $f: X \to P := \mathbb{P}_k^n$ a closed immersion. (And identify the set of hyperplanes of $P$ with $P' := \mathbb{P}_k^n(k)$)
Then for almost all hyperplanes $H$ in $P$, $X \not\subseteq H$ and $H \cap X$ is irreducible.
("almost all" means "there exists an open subset $U$ of $P'$ and for all $H \in U(k)$".)
I know this is theorem 6.3 of Jouanolou's "Theoremes de Bertini et applications".
But its proof is very difficult for me because it is too long, too general and written in French...
The author shows it under the hypothesis that $k$ is arbitrary infinite field, $X$ is just a geometrically irreducible $k$-shceme and $f$ is just a $k$-morphism.
So I'm glad if there are some short proofs of Bertini in my situation.
If it becomes easier, I'm willing to restrict the hypothesis to normal $X$.
This statement is very common (e.g., is used in Mumford's Abelian varieies), but I don't know any good scheme-theoritic proofs.
I have tried it by mimicking the proof of Hartshorne. However its proof hevily relies on the regularity of $X$, and so I couldn't.
So my question is: Would you show Bertini under the highlighted hypothesis? Or would you give some references which include a short proof of Bertini?
Thank you very much!