Bijective Cremona transformations of the plane

Abstract

We study Cremona transformations that induce bijections on the k-rational points. These form a subgroup inside the Cremona group. When k is a finite field, we study the possible permutations induced on P 2 (k), with special attention to the case of characteristic two. IntroductionWe call a birational self-map of a variety a birational permutation if both it and its inverse are defined on all rational points of the variety. In particular, such a map induces a bijection on the set of rational points. Over a finite field, the rational points form a finite set, so such bijective maps induce permutations in the usual sense. Fixing a variety and a finite ground field, one can ask what kind of permutations on the rational points can be realized this way.

Cremona groups over finite fields, Neretin groups, and non-positively curved cube complexes

“…Remark 1.6. The fact that if |k| = 2 n ≥ 4, then the permutations on the k-rational points P 2 k (k) induced by elements from BBir(P 2 k ) are all even, was conjectured in the preprint [ALNZ19]. We learned that in parallel to our work, Asgarli, Lai, Nakahara, and Zimmermann came up independently of us with a different proof of this conjecture, which they are going to include in a revised version of the preprint [ALNZ19].…”

“…The fact that if |k| = 2 n ≥ 4, then the permutations on the k-rational points P 2 k (k) induced by elements from BBir(P 2 k ) are all even, was conjectured in the preprint [ALNZ19]. We learned that in parallel to our work, Asgarli, Lai, Nakahara, and Zimmermann came up independently of us with a different proof of this conjecture, which they are going to include in a revised version of the preprint [ALNZ19]. Their methods are different from ours: they study the group BBir(P 2 k ) in detail, provide an explicit generating set, and show by hand that every element in the generating set induces an even permutation.…”

“…In this article, we will focus on Cremona groups Bir(P 2 k ) over finite fields k -a subject that has attracted substantial interest recently, also because of its connections to cryptography (see for instance [SB21], [Sch20], [SZ20], [ALNZ19], [LS21]). One of the reasons why birational transformations of surfaces are well understood, is the fact that they can be factorized into a sequence of blow-ups of points and contractions of curves.…”

“…Their methods are different from ours: they study the group BBir(P 2 k ) in detail, provide an explicit generating set, and show by hand that every element in the generating set induces an even permutation. In our paper, we use a result (Theorem 3.14) from [ALNZ19]. However, our main strategy for the proof of Theorem 1.5 is different and our result is more general, since it shows that the permutations of S(k) induced by BBir(S) are even for arbitrary regular rational projective surfaces S.…”