论文标题
Severi-Brauer表面的异性图，并应用于较高等级的Cremona组
Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank
论文作者
论文摘要
我们证明，$ \ mathbb {c} $中的任何一组基数都是任何cremona排名的商，至少$ 4 $。这为Cremona群体的商人的问题提供了明确的答案。结果，这给出了I. Dolgachev的问题的负面答案，即Dolgachev是否通过参与产生了所有等级的Cremona群体。作为另一个应用程序，我们表明，更高的Cremona群体不享受某些经典的群体理论属性（即霍普夫属性物业），这些属性是由$ 2 $的Cremona Groups满足的。最后，我们发现，Cremona排名的$ 3 $转折至少$ 4 $是不可数的。 为了推断高级克雷莫纳群体的这些特性，我们首先描述了一个非平凡的severi-brauer表面$ s $在一个完美领域上的异性转变，特别是证明，如果$ s $包含一个学位$ 6 $，那么它的Birational自图群，那么它的Birational Selftrational Maps就不会由有限的命令来录制为quemormormphism $ susefive $ $ nausefive $ $ suseftive $ $ nabistive $ suseftive $ suseftive $。然后，我们使用此结果来研究复数领域的Mori纤维空间，该纤维是非平凡的透明表面。
We prove that any group of cardinality at most the one of $\mathbb{C}$ is a quotient of any Cremona group of rank at least $4$. This provides a definitive answer to the question of what the quotients of Cremona groups can be. As a consequence, this gives a negative answer to the question of I. Dolgachev of whether Cremona groups of all ranks are generated by involutions. As another application, we show that higher Cremona groups do not enjoy some classical group-theoretic properties (namely, the Hopfian property) which are satisfied by Cremona groups of rank $2$. Finally, we discover that the $3$-torsion of the Cremona group of rank at least $4$ is not countable. To deduce these properties of higher Cremona groups, we first describe the group of birational transformations of a non-trivial Severi-Brauer surface $S$ over a perfect field, proving in particular that if $S$ contains a point of degree $6$, then its group of birational self-maps is not generated by elements of finite order as it admits a surjective group homomorphism to~$\mathbb{Z}$. We then use this result to study Mori fibre spaces over the field of complex numbers, for which the generic fibre is a non-trivial Severi-Brauer surface.
