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I'm supposed to show that for all genuses $g\geq3$, not all curves are hyperelliptic. Let me preface this by saying that I am not taking a what I think probably is a typical algebraic geometry course. We are apparently studying Moduli Spaces of Curves $M_{g,n}$, homology, and Ribbon Graphs. This is an "undergraduate seminar" where by the end of the course the idea is that we will have gotten a glimpse at a sliver of modern research mathematics. So I don't really have much background in algebraic geometry.

I'm supposed to show this by comparing the dimension of hyperelliptic curves and the dimension of $M_{g,0}$, and using the informally stated fact that if a hyperellitpic curve of genus $g$ admits a map to $\mathbb{P}^1$ which makes it into a double cover branched over a number of points, such a map is unique. This would show that the space of hyperelliptic curves of genus $g\geq2$ is isomorphic to $M_{0,k}$ mod $\Sigma_k$, where I know $k$ is the number of points in relation to the genus, which is $2g+2$ by using the Euler Characteristic.

I have in my notes $dim_{\mathbb{C}}(M_{g,n})=3g-3+n$ except for a few cases. I think a hyperelliptic curve is something like $y^2=$ (polynomial of degree $2g+2$). This question really confuses me. I have no idea about the dimension of a hyperelliptic curve, except that I know an elliptic curve is like a torus, and somehow elliptic curves are like $M_{1,1}$ because any three points can be sent to zero, one, and infinity by a Mobius Transformation and then the last point parametrizes the space or something. If anyone can help me understand anything and explain things I that would be very helpful. I'm sort of lost.

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    $\begingroup$ I think your question already contains almost everything you need. On the one hand, hyperelliptic curves of genus $g$ are parametrised by a choice of $2g+2$ points in $\mathbf P^1$, so their moduli space has dimension at most $2g+2$. On the other hand, the moduli space of curves of genus $g$ has dimension $3g-3$. This already shows that for $g \geq 5$, not every genus $g$ curve is hyperelliptic. To finish the argument for $g=3$ and $g=4$, notice that if two sets of $2g+2$ points in $\mathbf P^1$ are in the same orbit of $PGL(2)$, then the corresponding double covers will be isomorphic. So... $\endgroup$ – Asal Beag Dubh Feb 20 at 9:36
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    $\begingroup$ in fact the dimension of the space of hyperelliptic curves of genus $g$ is not $2g+2$ but rather $2g+2 - \operatorname{dim} PGL(2)$. I'll let you finish from here.. $\endgroup$ – Asal Beag Dubh Feb 20 at 9:38

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