TITLE: Bertini's theorem over non-algebraically closed field
QUESTION [7 upvotes]: Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a general element of the linear system $|\mathcal{L}|$ is a smooth $K$-variety? If not true in general, is there any condition on $X$ under which this holds true?

REPLY [11 votes]: This is true both over finite and infinite fields.

For infinite fields, see [Jou, Cor. I.6.11(2)]. It works for a general section of any very ample line bundle $\mathscr L$, using that over an infinite field a nonempty open subset of $|\mathscr L| \cong \mathbb P^N$ contains a rational point.
For finite fields, see [Poo, Thm. 1.1]. It requires taking a high power of your given very ample line bundle $\mathscr L$.


References.
[Jou] Jouanolou, Jean-Pierre, Théorèmes de Bertini et applications. Progress in Mathematics, 42. Birkhäuser, Boston-Basel-Stuttgart (1983). ZBL0519.14002.
[Poo] Poonen, Bjorn, Bertini theorems over finite fields. Ann. Math. (2) 160.3 (2005), p. 1099-1127. ZBL1084.14026.