MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

Here is an example of a birational involution of $\mathbb P^n$ that is at least not obviously a combination of Cremona transformations:
Suppose $H\subset \mathbb P^n$ P^{n+1}$is a hypersurface of degree$d$in$\mathbb P^{n+1}$such that it contains two points$P_1,P_2\in H$such that the multiplicity of$H$at$P_i$is exactly$d-1$for$i=1,2$. Such hypersurfaces can be constructed with a little work. Now the projection from$P_i$defines a birational isomorphism between$H$and$\mathbb P^n$and the composition of these defines a birational involution on$\mathbb P^n$. 1 Here is an example of a birational involution of$\mathbb P^n$that is at least not obviously a combination of Cremona transformations: Suppose$H\subset \mathbb P^n$is a hypersurface of degree$d$in$\mathbb P^{n+1}$such that it contains two points$P_1,P_2\in H$such that the multiplicity of$H$at$P_i$is exactly$d-1$for$i=1,2$. Such hypersurfaces can be constructed with a little work. Now the projection from$P_i$defines a birational isomorphism between$H$and$\mathbb P^n$and the composition of these defines a birational involution on$\mathbb P^n\$.