Let $\mathbf{ K}$ be an algebraically closed field. The Cremona group $\operatorname{ Cr}_2(\mathbf{ K})$ is the group of birational transformations of the projective plane $\mathbb{ P}^2_{ \mathbf{ K}}$. We carry out an overall study of centralizers of elements of infinite order in $\operatorname{ Cr}_2(\mathbf{ K})$ which leads to a classification of embeddings of $\mathbf{ Z}^2$ into $\operatorname{ Cr}_2(\mathbf{ K})$, as well as a classification of maximal non-torsion abelian subgroups of $\operatorname{ Cr}_2(\mathbf{ K})$ when $\operatorname{ char}(\mathbf{ K})=0$.
Categories: math.AG