Faithful actions of automorphism groups of free groups on algebraic varieties

Considering a certain construction of algebraic varieties $X$ endowed with an algebraic action of the group $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$, $n<\mathrm{\infty }$, we obtain a criterion for the faithfulness of this action. It gives an infinite family $\mathcal{F}$ of $X$’s such that $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$ embeds into $\mathrm{A}\mathrm{u}\mathrm{t}(X)$. For $n⩾3$, this implies nonlinearity, and for $n⩾2$, the existence of $F_2$ in $\mathrm{A}\mathrm{u}\mathrm{t}(X)$ (hence nonamenability of the latter) for $X\in \mathcal{F}$. We find in $\mathcal{F}$ two infinite subfamilies $\mathcal{N}$ and $\mathcal{R}$ consisting of irreducible affine varieties such that every $X\in \mathcal{N}$ is nonrational (and even not stably rational), while every $X\in \mathcal{R}$ is rational and $3n$-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties $Z$, for which $\mathrm{A}\mathrm{u}\mathrm{t}(Z)$ contains the braid group $B_n$ on $n⩾3$ strands, does not exceed $3n$. This upper bound strengthens the one following from the paper by D. Krammer [Kr02], where the linearity of $B_n$ was proved (this latter bound is quadratic in $n$). The same upper bound also holds for $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$. In particular, it shows that the minimal rank of the Cremona groups containing $\mathrm{A}\mathrm{u}\mathrm{t}(F_n)$, does not exceed $3n$, and the same is true for $B_n$ if $n⩾3$. This paper is a major revision of [Po21].