Canonical scroll of a trigonal curve ($C\in |3M+(4-g)L|$)

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 $a\equiv g(\mathrm{mod}2)$.