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!