TITLE: Does a birational involution of C^n always have a fixed point?
QUESTION [7 upvotes]: The title describes completely the question.
For n=1 it is an easy exercise. For n=2 the statement is still true, and depends on the classification of birational involutions of P^2 (see e.g. http://arxiv.org/abs/math.AG/9907028). For n=3 and above I could not find any useful hint in the literature.

REPLY [3 votes]: The answer is no. There are plenty of counterexamples, for example the map given by "a-fortiori", which is $(x,y)\mapsto (x+1/y,-y)$, which can be generalised to any dimension.
The good question is probably to look at birational involutions of $\mathbb{P}^n$.