Extending self-maps to projective space over finite fields

Abstract

Using the closed point sieve, we extend to finite fields the following theorem proved by A. Bhatnagar and L. Szpiro over infinite fields: if $X$ is a closed subscheme of ${P}^n$ over a field, and $\phi \colon X \to X$ satisfies $\phi^* \mathscr{O}_X(1) \isom \mathscr{O}_X(d)$ for some $d \ge 2$, then there exists $r \ge 1$ such that $\phi^r$ extends to a morphism ${P}^n \to {P}^n$.