Finite abelian subgroups of the Cremona group of the plane

TL;DR

This paper classifies finite abelian subgroups of the Cremona group of the plane by analyzing automorphism groups of Del Pezzo surfaces and conic bundles, revealing new conjugacy class structures and properties of elements.

Contribution

It provides a comprehensive classification of finite abelian subgroups in the Cremona group, including conjugacy invariants and the existence of infinitely many classes for certain orders.

Findings

Classification of conjugacy classes of finite abelian subgroups

Existence of infinitely many conjugacy classes of elements of even order

Linear transformations of finite order are conjugate to linear transformations

Abstract

This work presents the conjugacy classes of finite abelian subgroups of the Cremona group of the plane. Using a well-known theory, this problem amounts to the study of automorphism groups of some Del Pezzo surfaces and conic bundles. We have thus to enumerate all the cases, which gives a long description, and then to show whether two cases are distinct or not, using some conjugacy invariants. For example, we use the non-rational curves fixed by one element of the group, and the action of the whole group on these curves. From this classification, we deduce a sequence of more general results on birational transformations, as for example the existence of infinitely many conjugacy classes of elements of order n, for any even number n, a result false in the odd case. We prove also that a root of some linear transformation of finite order is itself conjugate to a linear transformation.