6
$\begingroup$

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?

$\endgroup$
6
$\begingroup$

(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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.