ag.algebraic geometry - Does a birational involution of C^n always have a fixed point?

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 $mathbb{P}^n$.