论文标题
自由构成群体的忠实行动在代数品种上
Faithful actions of automorphism groups of free groups on algebraic varieties
论文作者
论文摘要
考虑到一定的代数品种$ x $赋予了该组$ {\ rm aut}（f_n）$，$ n <\ infty $的代数行动的$ x $，我们获得了该动作忠诚的标准。它给出了$ x $的无限族$ \ mathscr f $ f $ of $ x $的s，使得$ {\ rm aut}（f_n）$嵌入到$ {\ rm aut}（x）$中。对于$ n \ geqslant 3 $，这意味着非线性，对于$ n \ geqslant 2 $，存在$ f_2 $ in $ {\ rm aut}（x）$中的$ f_2 $（因此，后者对后者的性不可能），对于$ x \ in \ mathscr f $。我们在$ {\ mathscr f}中找到两个无限的亚家族$ {\ mathscr n} $和$ \ mathscr r $由不可约的仿射品种组成，因此每个$ x \ in {\ mathscr n}中的每个$ x \ in {\ mathscr n} $ intrational（甚至不是稳定的）$ x \ in $ x \ in $ x \ ins $ x \ in cr and $ x \ in cr cr。作为一个应用程序，我们表明仿射代数品种$ z $的最小尺寸，$ {\ rm aut}（z）$包含$ n \ geqslant 3 $ strands上的辫子组$ b_n $，不超过$ 3n $。该上限加强了D. Krammer [KR02]的论文中的以下内容，在该论文中，证明了$ b_n $的线性（后者绑定在$ n $中是二次的）。相同的上限也适用于$ {\ rm aut}（f_n）$。特别是，它表明包含$ {\ rm aut}（f_n）$的Cremona组的最低等级不超过$ 3N $，如果$ n \ geqslant 3 $，则$ b_n $也是如此。 本文是[PO21]的主要修订版。
Considering a certain construction of algebraic varieties $X$ endowed with an algebraic action of the group ${\rm Aut}(F_n)$, $n<\infty$, we obtain a criterion for the faithfulness of this action. It gives an infinite family $\mathscr F$ of $X$'s such that ${\rm Aut}(F_n)$ embeds into ${\rm Aut}(X)$. For $n\geqslant 3$, this implies nonlinearity, and for $n\geqslant 2$, the existence of $F_2$ in ${\rm Aut}(X)$ (hence nonamenability of the latter) for $X\in \mathscr F$. We find in ${\mathscr F}$ two infinite subfamilies ${\mathscr N}$ and $\mathscr R$ consisting of irreducible affine varieties such that every $X\in {\mathscr N}$ is nonrational (and even not stably rational), while every $X\in \mathscr R$ is rational and $3n$-dimensional. As an application, we show that the minimal dimension of affine algebraic varieties $Z$, for which ${\rm Aut}(Z)$ contains the braid group $B_n$ on $n\geqslant 3$ 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 ${\rm Aut}(F_n)$. In particular, it shows that the minimal rank of the Cremona groups containing ${\rm Aut}(F_n)$, does not exceed $3n$, and the same is true for $B_n$ if $n\geqslant 3$. This paper is a major revision of [Po21].
