Curves of genus 𝑔 whose canonical model lies on a surface of degree 𝑔+1

Abstract

Let C C be a non-hyperelliptic curve of genus g g . We prove that if the minimal degree of a surface containing the canonical model of C C in P ˇ k g − 1 \check {\mathbb {P}}_k^{g-1} is g + 1 g+1 , then either g ≥ 9 g\ge 9 and C C carries exactly one g 4 1 g^{1}_{4} or 7 ≤ g ≤ 15 7\le g\le 15 and C C is birationally isomorphic to a plane septic curve with at most double points as singularities.