TITLE: Geometrically rational variety over a finite field
QUESTION [6 upvotes]: Let $k=\mathbb{F}_q$ be a finite field, and let $X$ be a smooth projective variety over $k$. Suppose that $X_{\overline{k}}$ is birational to $\mathbb{P}^n_{\overline{k}}$, do we know
(1)If $X$ is necessarily birational to $\mathbb{P}^n_k$?
(2)If $X$ necessarily has a $k$-point?

REPLY [6 votes]: (1) No: There exist minimal cubic surfaces over finite fields (see for example https://arxiv.org/abs/1611.02475). Such surfaces are non-rational over the ground field.
(2) Yes: This is a special case of a more general result of Esnault: https://arxiv.org/abs/math/0207022
This proves the congruence $\#X(\mathbb{F}_q) \equiv 1 \bmod q$, which clearly implies the existence of a rational point.