 Trigonal curve

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.