Extending Self-Maps to Projective Space over Finite Fields

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 $\{P}^n \to \{P}^n$.

2010 Mathematics Subject Classification: Primary 37P25; Secondary 37P55

Keywords and Phrases: self-map, closed point sieve

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