论文标题
连接的代数组，作用于$ \ mathbb {p}^1 $上的FANO纤维
Connected algebraic groups acting on Fano fibrations over $\mathbb{P}^1$
论文作者
论文摘要
令$ x/\ mathbb {p}^1 $为Mori纤维空间，Picard的一般光纤至少两个。我们证明，有一个适当的封闭子集$ s \ subsetNeq x $，由身份$ {\ rm aut}^{\ circ}^{\ circ}（x）$ $ x $的连接组件不变，这是$ x $ aut}^{\ circ}（x）$在$ \ mathbb {p}^1 $上琐碎地行动。这样的结果是一种将均等的birational图描述为从$ x/\ mathbb {p}^1 $到其他Mori纤维空间的工具，因此在研究相关的代数亚组的研究中找到了其应用。这代表了朝着可能分类的第一步，可以对Cremona组的最大连接代数子组进行分类，$ 4 $。
Let $X/\mathbb{P}^1$ be a Mori fibre space with general fibre of Picard rank at least two. We prove that there is a proper closed subset $S\subsetneq X$, invariant by the connected component of the identity ${\rm Aut}^{\circ}(X)$ of the automorphism group of $X$, which is moreover the orbit of a section $s$ and whose intersection with a fibre is an orbit of the subgroup of ${\rm Aut}^{\circ}(X)$ acting trivially on $\mathbb{P}^1$. Such result is a tool to describe equivariant birational maps from $X/\mathbb{P}^1$ to other Mori fibre spaces and therefore finds its applications in the study of connected algebraic subgroups of ${\rm Aut}^{\circ}(X)$. This represents a first reduction step towards a possible classification of maximal connected algebraic subgroups of the Cremona group of rank $4$.
