Every smooth projective curve embeds in \(\PP^3\), and is birational to a plane curve with at worst nodes.
(The projection criterion)
Let \(C \subseteq \PP^N\) and \(O \notin C\), and let \(\varphi\) be projection away from \(O\). Then \(\varphi\) is a closed immersion exactly when \(O\) lies on no secant and no tangent line of \(C\). The secant variety has dimension at most \(3\) and the tangent variety at most \(2\), so for \(N \geq 4\) such a point \(O\) exists. For \(N=3\), \(\varphi\) is birational onto a nodal image exactly when \(O\) lies on only finitely many secants, on no tangent, on no multisecant, and on no secant whose two tangent lines are coplanar.
The dimension count is the whole argument and is the thing to say out loud: secants sweep at most a threefold, tangents at most a surface, and \(\PP^N\) has room to avoid both as soon as \(N \geq 4\). Projection then drops the ambient dimension one step at a time until it stops at \(\PP^3\), where the secant variety fills the space and further projection must lose injectivity.
The consequence for genus is the nodal formula: if a plane curve of degree \(d\) has \(r\) nodes and nothing worse, its normalization has genus \[ g = \binom{d-1}{2} - r . \] So classifying all curves reduces to studying the family of plane curves of degree \(d\) with \(r\) nodes, which is nonempty exactly for \(0 \leq r \leq \binom{d-1}{2}\) and, by Severi and Harris, irreducible of dimension \(\tfrac{1}{2}d(d+3) - r\). The two extremes are Bertini at \(r=0\) and, at the top, the rational curve obtained by projecting the degree-\(d\) rational normal curve in \(\PP^d\) down to the plane.
Dependencies
PR-CRVDEGBDThe degree thresholds $2g$ and $2g+1$usesD-G1AEHThe genus in its several sensesuses
Backlinks
D-CRVPLSINGNode, cusp, and tacnode; the delta invariant and the genus formularelated-to