Canonical curves on surfaces of very low degree

Abstract:

Let $C$ be a non–hyperelliptic curve of genus $g$. We recall some facts about curves endowed with a base–point–free $g^{1}_{4}$. Then we prove that if the minimal degree of a surface containing the canonical model of $C$ in $\check {\mathbb {P}}^{g-1}_k$ is $g$, then $7\le g\le 12$ and $C$ carries exactly one $g^{1}_{4}$. As a by–product, we deduce that if the canonical model of $C$ in $\check {\mathbb {P}}^{g-1}_k$ is contained in a surface of degree at most $g$, then $C$ is either trigonal or tetragonal or isomorphic to a plane sextic.