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 $\mathbb{P}_k^2$?