A remark on polycyclic groups of birational automorphisms

Abstract

Let $X$ be an algebraic variety of dimension $d$ over an algebraically closed field $k$ of zero characteristic. Suppose that $G$ is a virtually polycyclic subgroup in the group $\mathrm{Bir}(X)$ of birational automorphisms of $X$. We show that the virtual derived length of $G$ does not exceed $2d+1$. Moreover, if $X$ is a surface, the bound can be improved to $3$, and this value is optimal.