By a *variety* over a field $k$, I mean a scheme that is separated and
of finite type over $k$. I indicate changes of the base ring by
subscripts.

Does there exist a smooth and projective variety $V$ over some field $k$, with $V(k)\neq\emptyset$, that isgeometrically unirational, i.e., there exists a dominant rational map $$\mathbf{P}^n_{\overline{k}} \dashrightarrow V_{\overline{k}}$$ for some integer $n$, but not unirational, i.e., there doesnotexist a dominant rational map $$\mathbf{P}^n_k \dashrightarrow V$$ for any integer $n$.

I don't recall ever coming across an example of a $V$ as in the question. It is a classical fact that any such $V$ has to have dimension at least $2$. Furthermore, I think I would have known if any examples existed where the dimension of $V$ is $2$. On the other hand, while the condition $V(k)\neq\emptyset$ is certainly needed to keep the question from having a trivial answer in the positive (e.g., let $V$ be a smooth cubic surface in $\mathbf{P}^3_k$ that does not have a $k$-point), it seems too unnatural to me to be sufficient for a negative answer to the question in general.

NB. I realize that some people (including myself at times) use the term "(uni)rational" for the concept I'm referring to as "geometrically (uni)rational". However, when comparing the two properties, I somehow prefer talking about "geometrically unirational vs. unirational" to "unirational vs. unirational over the ground field".

**Some edits made** to reflect comments by Jason Starr and ayanta; see below.

`$V:=\{y^q=x-tx^p\}$`

for a $p$-power $q > 2$. This is a smooth irreducible $k$-subgroup of $\mathbf{G}_a^2$ with closure in $\mathbf{P}^2_k$ that is regular (butnot $k$-smooth). Thus, $V_{\overline{k}} \simeq \mathbf{G}_a$. But $V$ is not unirational over $k$ since $V(k)$ is finite and hence not Zariski-dense in $k$: if $p > 2$ then`$V(k) = \{(0,0)\}$`

, whereas if $p=2$ then`$V(k)=\{(0,0),(1/t,0)\}$`

since $q > 2$ (for $q=2$ it is a smooth affine conic with a $k$-point, so rational!). $\endgroup$6more comments