Let $k$ be an algebraically closed field. Let $X$ be a smooth, projective variety over $k$ that is separably rationally connected, i.e., there exists a $k$-morphism $u:\mathbb{P}^1_k \to X$ such that $u^*T_X$ is isomorphic to $\mathcal{O}_{\mathbb{P}^1_k}(a_1)\oplus \dots \oplus \mathcal{O}_{\mathbb{P}^1_k}(a_n)$ for positive integers $a_1,\dots,a_n$. Let $f:X\to X$ be a $k$-automorphism.

**Question**. Does there exist a $k$-point of $X$ that is fixed by $f$?

This is true if $k$ is of characteristic $0$ by the Atiyah-Bott fixed point theorem. That theorem was extended to positive characteristic by many authors; one reference is Corollaire 6.12 (Appendice), Exposé III, SGA 5. This extension does not quite give the result above; it does give the result if $h^i(X,\mathcal{O}_X)$ vanishes for all $i>0$, but that is unknown for arbitrary separably rationally connected, smooth, projective varieties.

By a theorem of Kollár, this is also true in positive characteristic if $f$ has finite order. By the "spreading out" technique, this implies the result if there exists an ample invertible sheaf $\mathcal{L}$ such that $f^*\mathcal{L}$ is isomorphic to $\mathcal{L}$, i.e., if for some $n>0$, the iterate $f^n$ is in the identity component of the automorphism group scheme of $X$. In particular, the result is true if $X$ is separably rationally connected and Fano. However, there do exist pairs $(X,f)$ with $X$ a separably rationally connected variety and $f$ an automorphism that preserves no ample divisor class, e.g., translations on the elliptic surface obtained from $\mathbb{P}^2$ by blowing up the base locus of a pencil of plane cubics.

This question is related to the following questions: let $p$ be a prime integer, let $q$ be $p^r$, and let $Y/\mathbb{F}_q$ be a smooth, projective variety whose base change to $\overline{\mathbb{F}}_q$ is separably rationally connected. Let $g:Y \to Y$ be an $\mathbb{F}_q$-automorphism. What can we say about the induced permutation of the finite set $Y(\mathbb{F}_q)$ (whose cardinality is congruent to $1$ modulo $q$, by work of Esnault)? Does there exist an integer $N$ depending only on $p$ (not $q$) and geometric properties of $Y$ and $g$ such that there exists an orbit of size $\leq N$?