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?



What is known about the birational involutions of P^3?
Let $\mbox{Inv}(\mathbb P^n)$ be the 
 subgroup of $\mbox{Bir}(\mathbb{P}^n)$ generated by involutions and 
 $\mbox{PGL}(n+1, \mathbb C)$.



> 
> If $n\ge 3$ then any generating set of $\mbox{Inv}(\mathbb P^n)$
>  contains uncountably many involutions. More precisely
>  for any $d >1$ there are 
>  uncountably many
>  involutions of degree $\ge d$ in any generating set of $\mbox{Inv}(\mathbb P^n)$. 
> 
> 
> 


This is a direct consequece of the proof of [Hudson-Pan Theorem](http://www.springerlink.com/content/j6034487lw5h4334/) which asserts that $\mbox{Bir}(\mathbb{P}^n)$, $n\ge 3$, needs uncountably many generators of degree greater than $d$ for any $d>1$.


Let me briefly review Pan's proof adapting it to prove the remark above. It is achivied in two steps:


1. for each hypersurface $H$ of $\mathbb P^{n-1}$ there is a birational transformation of $\mathbb P^n$ which contracts a cone over $H$;
2. a hypersurface contracted by a product of birational transformations $f\_1 \circ f\_2 \circ \cdots \circ f\_k$ is birational to an hypersurface contracted by one of the birational transformations $f\_1, \ldots, f\_k$.


The proof of 2 is straight-forward. To prove 1, fix $p \in \mathbb P^n$ and consider the subgroup $\mbox{St}\_p(\mathbb P^n) \subset \mbox{Bir}(\mathbb P^n)$ which sends
lines through $p$ to lines through $p$. One can show that $\mbox{St}\_p(\mathbb P^n)$
fits into the split exact sequence
$$
1 \to \mbox{PGL}(2,\mathbb C(\mathbb P^{n-1})) \to \mbox{St}\_p(\mathbb P^n) \to \mbox{Bir}(\mathbb P^{n-1}) \to 1 
$$
where the rightmost map is defined by the action on the space of lines through $p$, and the leftmost is defined by the action on the fibers of the $\mathbb P^1$-bundle obtained from $\mathbb P^n$ after blowing up $p$. 


Now given $h \in \mathbb C(x\_0, \ldots, x\_{n-1})$, we can consider the element of 
$\mbox{PGL}(2,\mathbb C(\mathbb P^{n-1}))$ defined by 
$$ 
(s:t) \mapsto (t:h\cdot s) \, . 
$$
Clearly, this defines a birational involution of $\mathbb P^{n-1} \times \mathbb P^1$
which contracts the divisor associated to $h$. Of course, we can realize it as an element
of $\mbox{St}\_p(\mathbb P^n)$ inducing the identity on $\mbox{Bir}(\mathbb P^{n-1})$ and 
contracting a cone over $H=h^{-1}(0)$ with vertex at $p$. 


Notice that we can choose uncountably many $H$ in uncountably many distinct birational
equivalence classes. Putting this observation together with 1 and 2 allow us to conclude. 




---


You are probably
aware of it, but there is classification of birational involutions of $\mathbb P^2$ by
[Bayle-Beauville](http://arxiv.org/abs/math/9907028) which completes and clarifies previous works by Bertini and others.
Composing the involutions of $\mathbb P^2$ with elements of $PGL(2,\mathbb C(\mathbb P^2))$
gives many examples of birational involutions of $\mathbb P^3$ which are also in $\mbox{St}\_p(\mathbb P^3)$.


For more on $\mbox{St}\_p(\mathbb P^n)$ see this [other paper](http://www.ams.org/journals/proc/2001-129-05/S0002-9939-00-05749-X/home.html) of Pan. 


