$p$-elementary non-cyclic subgroups of the Cremona group of the plane

Abstract

We classify, up to conjugacy, the subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r \geq 2$, over an algebraically closed field $\mathbf{k}$ of characteristic not equal to $p$. In particular, we show that $r \leq 2$ if $p \geq 5$, $r \leq 3$ if $p=3$, and $r \leq 4$ if $p=2$. And hence reprove a well-known statement contained in the article of Arnaud Beauville "$p$-elementary subgroups of the Cremona group", 2007. The main contribution of this article is a concrete description of these groups. In fact, we give an explicit list of representatives via a set of 20 families consisting of subgroups of the de Jonqui\`eres group and subgroups of automorphisms of del Pezzo surfaces, and we study the possible conjugacies by birational maps between these families. Finally, we give some results on subgroups of the Cremona group of the plane isomorphic to $(\mathbb{Z}/p\mathbb{Z})^r$, where $p$ is prime and $r$ is an integer, over an algebraically closed field $\mathbf{k}$ of characteristic equal to $p$.