We know from Hartshorne's "Algebraic geometry" that if a curve $C$ is smooth then there exists a closed immersion of $C$ into $\mathbb{P}^3$. Does this still hold if $C$ is not generically reduced or singular at finitely many points?



Can any curve be embedded into $\mathbb{P}^3$?
No. If a curve embeds into $\mathbb{P}^3$, its tangent space at every point has dimension $\leq 3$. This is a strong restriction on the possible singularities. For instance, 
a curve locally isomorphic to the union of the lines $x=y=z=0$, $x=y=t=0$, $x=z=t=0$, $y=z=t=0$ in $\mathbb{A}^4$ cannot be embedded in $\mathbb{P}^3$.


