Recall that a compact Riemann surface/algebraic curve $C$ is hyperelliptic if it admits a branched double cover $C \to \mathbb P^1$, where $\mathbb P^1$ is the complex projective line/Riemann sphere. Among those curves of hyperbolic type ($g \ge 2$), the only genus that admits such a double cover in general is $g = 2$. While this fact is essentially trivial from the perspective of Hartshorne's fourth chapter, I have always found this proof to feel like a nuclear flyswatter (at least over $\mathbb C$).
Intuitively, a hyperelliptic Riemann surface of genus $g > 2$ is one that can be "drawn conformally" (whatever exactly that may mean) in such a way that all $g$ of its holes are "lined up." As such, my intuition suggests that the proposition in the title is somehow analogous to the trivial statement from Euclidean plane geometry that every configuration of $n > 1$ points is collinear if and only if $n=2$, where the holes in the surface are treated as analogous to points in the plane. So my question is this: is there a "holomorphic geometric topological" proof of said proposition that proceeds roughly as described in this paragraph rather than using any big theorems like Riemann-Roch?
Best Answer
Here is a small variant on Eremenko's answer.
The "Fenchel-Nielsen" coordinates on the space of hyperbolic metrics on a surface $\Sigma_g$ can be described via a pants decomposition. This is a decomposition of the surface along a collection of curves, that split the surface into a union of disjoint $3$-punctured spheres.
Hyperbolic metrics on 3-punctured spheres making the boundary into totally geodesic curves are specified by the cuff lengths, i.e. three real parameters.
So if you have a surface $\Sigma_g$, take a pants decomposition. Notice that you can choose pants decomposition equivariant with respect to a hyperelliptic involution. So if you think about the constraints on Fenchel-Nielsen coordinates coming from the surface being hyperelliptic, these only occur for $g > 2$. With $g=2$ all the curves are preserved and pants exchanged via the hyperelliptic involution, i.e. there are no additional constraints to be hyperelliptic.
But when $g>2$, there are curves off the hyperelliptic axis in your pants decomposition, so there are cuff lengths that are constrained by the hyperelliptic involution.
This is a long version of my comment.