Trigonal curve
= Trigonal curve
{title2=$g^1_3$}
{wiki}
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.