# Extending self-maps to projective space over finite fields

### Bjorn Poonen

Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139-4307 USA

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

## Cite this article

Bjorn Poonen, Extending self-maps to projective space over finite fields. Doc. Math. 18 (2013), pp. 1039–1044

DOI 10.4171/DM/420