Abstract:
A year ago, the proof of the following theorem was discussed at a seminar. Let
$q=2^m$ and $q>=4$, then birational permutations of the projective plane induce
only even permutations of $\mathbb{F}_q$ - points of the projective plane. The idea of
the proof was to explicitly describe the generators of the group of birational
permutations, and prove for each generator that it induces an even permutation
of rational points.
This time we will prove the mentioned theorem in a completely different way.
Namely, following the article by A. Genevois, A. Lonjou and K. Urech, using
some technique, we generalize the concept of parity to the entire group of
birational automorphisms and show that all elements of finite order (in
particular, involutions) are even elements. We will also see that this
approach allows us to generalize the theorem to an arbitrary smooth rational
projective surface.
If there is time left, we will discuss what other statements and theorems can
be proved using a similar technique.
