On Elements of Prime Order in the Plane Cremona Group over a Perfect Field

Finite abelian subgroups of the Cremona group of the plane

The Number of Conjugacy Classes of Elements of the Cremona Group of Some given Finite Order

This note presents the study of the conjugacy classes of elements of some given finite order n in the Cremona group of the plane. In particular, it is shown that the number of conjugacy classes is infinite if n is even, n = 3 or n = 5, and that it is equal to 3 (respectively 9) if n = 9 (respectively 15), and is exactly 1 for all remaining odd orders. Some precise representative elements of the...

On the Inertia Group of Elliptic Curves in the Cremona Group of the Plane

We study the group of birational transformations of the plane that fix (each point of) a curve of geometric genus 1. A precise description of the finite elements is given; it is shown in particular that the order is at most 6, and that if the group contains a non-trivial torsion, the fixed curve is the image of a smooth cubic by a birational transformation of the plane. We show that for a smoot...

The Cremona group of the plane is compactly presented

This article shows that the Cremona group of the plane is compactly presented. To do this we prove that it is a generalised amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.

Elements and cyclic subgroups of finite order of the Cremona group

We give the classification of elements – respectively cyclic subgroups – of finite order of the Cremona group, up to conjugation. Natural parametrisations of conjugacy classes, related to fixed curves of positive genus, are provided.