当前位置: X-MOL 学术Acta Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Quotients of higher-dimensional Cremona groups
Acta Mathematica ( IF 3.7 ) Pub Date : 2021-06-01 , DOI: 10.4310/acta.2021.v226.n2.a1
Jérémy Blanc 1 , Stéphane Lamy 2 , Susanna Zimmermann 3
Affiliation  

We study large groups of birational transformations $\operatorname{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbf{C}$ or a subfield of $\mathbf{C}$. Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\operatorname{Bir}(X)$ is the Cremona group of rank $n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface. In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\operatorname{Bir}(X)$ to $\mathbf{Z}/2$, showing in particular that the group $\operatorname{Bir}(X)$ is not perfect, and thus not simple. As a consequence, we also obtain that the Cremona group of rank $n \geqslant 3$ is not generated by linear and Jonquières elements.

中文翻译:

高维克雷莫纳群的商

我们研究了大量的双有理变换 $\operatorname{Bir}(X)$,其中 $X$ 是至少 $3$ 的各种维度,定义在 $\mathbf{C}$ 或 $\mathbf{C 的子域上}$。两个突出的情况是当 $X$ 是射影空间 $\mathbb{P}^n$,在这种情况下 $\operatorname{Bir}(X)$ 是秩 $n$ 的克雷莫纳群,或者当 $X \子集 \mathbb{P}^{n+1}$ 是一个光滑的三次超曲面。在这两种情况下,更一般地,当 $X$ 是圆锥丛的双有理时,我们产生了从 $\operatorname{Bir}(X)$ 到 $\mathbf{Z}/2$ 的无限多个不同群同态,特别是组 $\operatorname{Bir}(X)$ 并不完美,因此并不简单。因此,我们还获得秩为 $n \geqslant 3$ 的 Cremona 群不是由线性和 Jonquières 元素生成的。
更新日期:2021-07-02
down
wechat
bug