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?



Bertini's theorem over non-algebraically closed field
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 $\mathscr L$.




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**References.**


[Jou] Jouanolou, Jean-Pierre, *Théorèmes de Bertini et applications*. Progress in Mathematics, **42**. Birkhäuser, Boston-Basel-Stuttgart (1983). [ZBL0519.14002](https://zbmath.org/?q=an:0519.14002).


[Poo] Poonen, Bjorn, [*Bertini theorems over finite fields*](http://dx.doi.org/10.4007/annals.2004.160.1099). Ann. Math. (2) 160.**3** (2005), p. 1099-1127. [ZBL1084.14026](https://zbmath.org/?q=an:1084.14026).


