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A curve is a smooth projective connected curve over an algebraically closed field.

Every curve of genus 2 is planar.

Also, every curve of genus 3 is planar.

But what about curves of genus 4?

What is the dimension of the subvariety defined by planar curves in the moduli space of genus $g$ curves?

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You seem to be under some misconception here: no smooth curve of genus $2$ is planar, actually!

Indeed, a plane curve has a degree $d$ and if it is smooth its genus is then $g=\frac {(d-1)(d-2)}{2}$.
So actually most smooth curves are non-planar because most integers are not of the form $\frac {(d-1)(d-2)}{2}$.
The smallest example is, as I said, $g=2$ but also smooth curves of genus $4,5,7,8,9,\ldots $ are all non-planar (this answers your question about $g=4$).

Also: all smooth plane curves of degree $4$ have indeed genus $3$, but some curves of genus $3$ are not planar, namely the hyperelliptic ones.

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  • $\begingroup$ Ow you're right! I was confused by the fact that we also write a hyperelliptic curve as $y^2 = f(x)$. So then maybe some singular model of our curve is probably always planar... $\endgroup$ – Avantquelle Jul 8 '12 at 13:27

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