Bijective Cremona transformations of the plane

TL;DR

This paper investigates which permutations of rational points on the projective plane over finite fields can be realized by birational self-maps, proving that no odd permutation occurs over certain fields, thus completing prior classification efforts.

Contribution

It establishes that no odd permutation arises from birational automorphisms over non-prime finite fields of characteristic two, advancing understanding of Cremona transformations.

Findings

No odd permutation over non-prime fields of characteristic two

Complete classification of permutations induced by Cremona transformations

Identification of generators for the group of birational self-maps

Abstract

We study the birational self-maps of the projective plane over finite fields that induce permutations on the set of rational points. As a main result, we prove that no odd permutation arises over a non-prime finite field of characteristic two, which completes the investigation initiated by Cantat about which permutations can be realized this way. Main ingredients in our proof include the invariance of parity under groupoid conjugations by birational maps, and a list of generators for the group of such maps.