TITLE: What is known about the birational involutions of P^3?
QUESTION [18 upvotes]: Describing the group of birational automorphisms of $\mathbb{P}^n$, $\mbox{Bir}(\mathbb{P}^n)$, for $n\ge 3$ is a fundamental open problem in birational geometry. For $n=2$, the classical theorem of Noether says that this group is generated by linear transformations and the Cremona transformation, which is given by
$$
\phi:(x_0 : x_1 : x_2) \to (x_0^{-1} : x_1^{-1} : x_2^{-1}).
$$For $n\ge 3$, there is an analogous Cremona transformation, but it is known that the group $\mbox{Bir}(\mathbb{P}^n)$ is no longer generated by this and $\mbox{PGL}_{n+1}(\mathbb{C})$. My question is therefore

Are there examples of other birational
  involutions of $\mathbb{P}^3$?

In case the answer is yes, have these been classified? I'm also interested in the analog of Noether's theorem in this case: Are there examples of birational transformations of $\mathbb{P}^3$ that can not be written as a composition of birational involutions?

REPLY [6 votes]: There are plenty of other examples of birational involutions of $\mathbb{P}^3$ which are known. For the moment, no precise classification as in dimension $2$. However, the following preprint of Yuri Prokhorov gives a first nice step toward a classification:
Yuri Prokhorov, "On birational involutions of $\mathbb{P}^3$"
http://arxiv.org/abs/1206.4985
In dimension $2$, we can say if two involutions are conjugate only by looking at fixed points. When the curves pointwise fix a curve of positive genus, the curve has to be the same (easy argument), but this condition is in fact sufficient (hard argument, only follows from classification). The same works for any order, looking carefully at the fixed points of all powers (see J. Blanc, "Elements and cyclic subgroups of finite order of the Cremona group."
Comment. Math. Helv. 86 (2011), no. 2, 469-497, http://arxiv.org/abs/0809.4673).
In dimension $3$, we can have the same discussion, replacing fixed curves of positive genus with fixed surfaces which are not birationally ruled (other surfaces can be contracted by birational maps). The article of Yuri Prokhorov studies the case where such fixed surface exists.