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

Casnati, Gianfranco

doi:10.1090/S0002-9939-2012-11335-8

Curves of genus $g$ whose canonical model lies on a surface of degree $g+1$
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Proc. Amer. Math. Soc. 141 (2013), 437-450 Request permission

Abstract:

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

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