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In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall that an involution on a complex manifold $X$ is an element $f \in \mathrm{Aut}(X)$ of order two. I want to prove the following results:

(i) The fixed point set of any involution on $\mathbb{P}^{2}$ contains a line.

(ii) Every involution on a non-hyperelliptic Riemann surface of genus 3 (i.e. a canonically embedded degree 4 curve) has a fixed point.

(iii) Every involution on a Riemann surface of even genus has a fixed point.

Could someone provide some guidance on this? Thanks very much!

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    $\begingroup$ If $\Sigma_{g'}$ is a double cover of $\Sigma_g$ then $g'=2g-1$, i.e. $g'$ is odd. This solves (iii). $\endgroup$ – user8268 Apr 14 '14 at 9:28
  • $\begingroup$ Thank you for your response. What exactly do you mean by the notation $\sum_{g^{\prime}}$? $\endgroup$ – user 3462 Apr 14 '14 at 16:21
  • $\begingroup$ $\Sigma_{g'}$ = a surface of genus $g'$; if you have an involution without fixed points on $\Sigma_{g'}$ then you get a 2-fold (unramified) covering $\Sigma_{g'}\to\Sigma_g$ for some $g$. $\endgroup$ – user8268 Apr 14 '14 at 21:48
  • $\begingroup$ Ah okay, that's what I figured, but I wanted to be sure. Any insight into the others? $\endgroup$ – user 3462 Apr 15 '14 at 3:55
  • $\begingroup$ For the first point, you could use the fact that every automorphism of $\mathbb{P}^2$ is linear. For the second point you can use again the strategy of user8268 and see what happens with the induced covering. $\endgroup$ – Daniele A Apr 15 '14 at 19:22

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