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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 $\mbox{PGL}_n(\mathbb{C})$ this and the Cremona transformation. $\mbox{PGL}_{n+1}(\mathbb{C})$. 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?

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# What is known about the birational involutions of P^3?

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 transformation, but it is known that the group $\mbox{Bir}(\mathbb{P}^n)$ is no longer generated by $\mbox{PGL}_n(\mathbb{C})$ and the Cremona transformation. 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?