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?



Geometrically rational variety over a finite field
(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.


