By degree, for smooth curves in \(\PP^3\):
\(d=1\): a line, \(g=0\).
\(d=2\): a plane conic, \(g=0\).
\(d=3\): the plane cubic with \(g=1\), and the twisted cubic with \(g=0\).
\(d=4\): the plane quartic with \(g=3\); the rational quartic with \(g=0\); and the elliptic quartic with \(g=1\), which is the complete intersection of two quadrics.
\(d=5\): the plane quintic with \(g=6\), and curves with nonspecial hyperplane section of genus \(0, 1, 2\).
\(d=6\): the plane sextic with \(g=10\); nonspecial curves of genus \(0,1,2,3\); and the canonical curve of genus \(4\), the complete intersection of a quadric and a cubic.
\(d=7\): the plane septic with \(g=15\); nonspecial curves of every genus \(g \leq 4\); curves of genus \(5\); and curves of genus \(6\), the maximum allowed by Castelnuovo’s bound, which all lie on a quadric, for instance the curves of type \((3,4)\) on a smooth quadric.
(Degree $7$ and genus $5$)
Let \(X\) have genus \(5\) and \(P \in X\). Since \(\abs{K}\) is base-point free, \(D = K - P\) has degree \(7\) and \(h^0(D) = 4\). By the very-ampleness criterion and Serre duality, \(D\) is very ample exactly when \(h^0(P + Q + R) = 1\) for all \(Q, R \in X\). If \(X\) has no \(g^1_3\) this holds for every \(P\), and \(\abs{K - P}\) embeds \(X\) in \(\PP^3\) as a curve of degree \(7\); curves of genus \(5\) without a \(g^1_3\) exist, so \((d, g) = (7, 5)\) occurs. The hyperplane section \(K - P\) is special, in accordance with \(g \geq \tfrac{1}{2} d + 1\). [@Har10a]
For \(g \geq 2\), a curve of genus \(g\) has a nonspecial very ample divisor of degree \(d\) exactly when \(d \geq g+3\), and then it embeds in \(\PP^3\). If the hyperplane section is special and \(C\) is not planar, then \(d \geq 6\) and \(g \geq \tfrac{1}{2}d + 1\); the only such curve with \(d=6\) is the canonical genus-\(4\) curve.
Two curves in \(\PP^3\) with the same \((d,g)\) need not be deformations of each other: for degree \(9\) and genus \(10\) there are complete intersections of two cubics and curves of type \((3,6)\) on a smooth quadric. They are distinguished by \(h^0(\mci_C(2))\), which is \(0\) for the first and \(1\) for the second. By upper semicontinuity of \(h^0(\mci_C(2))\) in flat families, a complete intersection of two cubics is not a flat limit of curves of type \((3,6)\).
Dependencies
T-CRVCASTCastelnuovo's bound for space curvesusesFE-CRVQUADCurves of type $(a,b)$ on a quadric surfaceuses