Let \(C \subseteq \PP^3\) be a smooth curve of degree \(d\) and genus \(g\) not contained in a plane. Then \(d \geq 3\) and \[ g \leq \begin{cases} \tfrac{1}{4}d^2 - d + 1 , & d \text{ even} , \\[2pt] \tfrac{1}{4}(d^2-1) - d + 1 , & d \text{ odd} . \end{cases} \] The bound is attained for every \(d \geq 3\), and a curve attaining it lies on a quadric surface.
The bound and its equality case are one statement: the extremal curves are the balanced curves on a quadric. Type \((a,a)\) on a smooth quadric has \(d = 2a\) and \(g = (a-1)^2 = \tfrac{1}{4}d^2 - d + 1\), and type \((a,a+1)\) has \(d = 2a+1\) and \(g = a^2 - a\), which is the odd formula. Thus the equality cases are exactly the balanced bidegrees on a quadric.
The contrast with plane curves explains the nonplanarity hypothesis. A plane curve of degree \(d\) has \(g = \binom{d-1}{2} \approx \tfrac{1}{2}d^2\), roughly twice Castelnuovo’s bound: spreading a curve out into \(\PP^3\) costs genus, and the hypothesis that \(C\) is not planar is what makes the smaller bound apply.
Dependencies
FE-CRVQUADCurves of type $(a,b)$ on a quadric surfaceuses
Backlinks
FE-CRVDEGSCurves of low degree in $\PP^3$uses