# Geometrically rational variety over a finite field

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?

This proves the congruence $$\#X(\mathbb{F}_q) \equiv 1 \bmod q$$, which clearly implies the existence of a rational point.