Trigonal curve ($g_3^1$)

A smooth projective curve of gonality three has a basepoint-free pencil of degree three. For a nonhyperelliptic curve, a fibre of this pencil spans a line in its canonical map: Riemann-Roch theorem gives $h^0(K_C-D)=g-2$. The union of these lines is a scroll of degree $g-2$ in ${\mathbb{P}}^{g-1}$. Its smooth ruled model may map to a cone in low genus.

The rank-two bundle of fibre spans is globally generated and splits as $\mathcal{O}(a_1)\oplus \mathcal{O}(a_2)$ with $a_1+a_2=g-2$. On its rational normal scroll, $L{|}_C$ is the degree-three pencil and $M{|}_C=K_C$. The adjunction formula then forces the displayed divisor class. Writing $a=a_2-a_1$ and $M=B+a_{2}L$ on the Hirzebruch surface gives $C\in |3B+(a+a_2+2)L|$.

The difference between the two normalized splitting degrees of the canonical scroll of a trigonal curve measures its imbalance. Since a smooth trigonal curve cannot contain the negative section, its intersection with that section is nonnegative. This gives $3a\le g+2$, and parity gives $a\equiv g(\mathrm{mod}2)$.