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7
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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. |
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2
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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 $\mathbb{P}^n$. |
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