Canonical scroll of a trigonal curve
= Canonical scroll of a trigonal curve
{title2=$C\in|3M+(4-g)L|$}
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 $M=B+a_2L$ on the <Hirzebruch surface> gives $C\in|3B+(a+a_2+2)L|$.