论文标题
仿射品种的自动形态何时线性？
When is the automorphism group of an affine variety linear?
论文作者
论文摘要
令$ aut_ {alg}（x）$是仿射代数品种$ x $的常规自动形态（x）$组的子组。我们证明，如果$ dim x \ ge 2 $，并且如果$ aut_ {alg}（x）$足够丰富，则$ aut_ {alg}（x）$不是线性的，即不能将其嵌入$ gl_n（k）$，其中$ k $是特征性零的代数封闭的字段。此外，$ AUT（X）$仅当$ AUT（X）$的连接组件是代数曲线或通勤单位群体的直接限制时，仅作为一个抽象组作为抽象组是同构。最后，我们证明，对于$ x $的无数$ k $，如果$ x $具有正维线性代数组的合理动作，则$ x $的偶然转型对仿射品种的自动形态不可能是同构。
Let $Aut_{alg}(X)$ be the subgroup of the group of regular automorphisms $Aut(X)$ of an affine algebraic variety $X$ generated by all connected algebraic subgroups. We prove that if $dim X \ge 2$ and if $Aut_{alg}(X)$ is rich enough, $Aut_{alg}(X)$ is not linear, i.e., it cannot be embedded into $GL_n(K)$, where $K$ is an algebraically closed field of characteristic zero. Moreover, $Aut(X)$ is isomorphic to an algebraic group as an abstract group only if the connected component of $Aut(X)$ is either the algebraic torus or a direct limit of commutative unipotent groups. Finally, we prove that for an uncountable $K$ the group of birational transformations of $X$ cannot be isomorphic to the group of automorphisms of an affine variety if $X$ is endowed with a rational action of a positive-dimensional linear algebraic group.
