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?
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5$\begingroup$ No. You always have a lower bound based on the dimension of the tangent space at any point. $\endgroup$– Anton FonarevJan 21, 2014 at 10:31
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1$\begingroup$ See also mathoverflow.net/a/14405 for smooth varieties over finite fields. $\endgroup$– CantlogJan 21, 2014 at 14:13
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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 $\mathbb{P}^3$.
