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In the chapter on curves in Hartshorne it is proved that every curve can be embedded in $\mathbb{P}_k^3$ and is birationally equivalent to a planar curve with at most nodes as singularities (Corollaries 3.6 and 3.11 of this chapter). Here a curve is by definition an integral scheme of dimension one which is proper and smooth over some algebraically closed field $k$.

Are there examples of curves which cannot be embedded in $\mathbb{P}_k^2$?

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up vote 8 down vote accepted

Hyperelliptic curves of any genus $\ge 2$ (e.g. curves of genus $2$) can't be embedded in the projective plane.

For curves of genus $2$, the proof is easy: plane curves of degree $d$ have genus $(d-1)(d-2)/2$, and this number can never be $2$.

Similarly, curves of genus 4, 5, 7 are never plane.

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