论文标题
关于克雷莫纳小组的三章
Three chapters on Cremona groups
论文作者
论文摘要
在本文的第一部分中，我们回答了I. Dolgachev的问题，该问题与以下问题有关：给定一个Birational Map $ f \ in \ Mathrm {bir}（\ Mathbb {p}^M_ \ Mathbf {K k}）$ $ a \ in \ mathrm {pgl} _ {m+1}（\ mathbf {k}）$，什么时候$ a \ circ f $ raranizizizable？ Dolgachev的最初问题是，这是否可能发生在$ \ Mathrm {pgl} _ {m+1}（\ Mathbf {k}）$中的所有$ A $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a {m+mathbf {k}）$中的最初问题，答案是负面的，答案是负面。然后，我们查看序列$ n \ mapsto \ mathrm {deg} \，f^n $，$ f \ in \ mathrm {bir}（\ mathbb {p}^2_ \ mathbf {k}）$。我们表明，$ n \ mapsto \ mathrm {deg} \，f^n- \ mathrm {deg} \，f^{n-1} $的序列没有约束。最终，我们研究了由Birational图不变的曲线铅笔程度。当F是Halphen或Jonquières的扭曲时，我们证明该度是由$ \ mathrm {deg} \，f $的函数所界的。我们在结合类的结构上得出了推论，以及它们相对于$ \ mathrm {bir}的Zariski拓扑的属性（\ Mathbb {p}^2_ \ Mathbf {k}）$。
In the first part of this article, we answer a question of I. Dolgachev, which is related to the following problem: given a birational map $f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map $A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable? Dolgachev's initial question is whether this may happen for all $A$ in $\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at the sequence $n\mapsto\mathrm{deg}\, f^n$, $f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-\mathrm{deg}\, f^{n-1}$ for small values of $n$. Finally we study the degree of pencils of curves which are invariant by a birational map. When f is a Halphen or Jonquières twist, we prove that this degree is bounded by a function of $\mathrm{deg}\, f$. We derive corollaries on the structure of conjugacy classes, and their properties with respect to the Zariski topology of $\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$.
