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Let $f:X\to Y$ be a non-constant morphism of smooth projective connected curves over $\mathbf{C}$ (or compact connected Riemann surfaces).

Suppose that $X=Y$ and that $f$ is not the identity.

Why are the fixed points of $f$ isolated?

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    @Ali: all you have to do is observe that $f(x) = x$ is an algebraic condition on $x$.2012-02-04

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