Exercise 111.56.9. Let $k$ be an algebraically closed field. Let $K/k$ be finitely generated field extension of transcendence degree $1$. Let $C$ be the abstract curve corresponding to $K$. Let $V \subset K$ be a $g^ r_ d$ and let $\Phi : C \to \mathbf{P}^ r$ be the corresponding morphism. Show that the image of $C$ is contained in a quadric^{1} if $V$ is a complete linear system and $d$ is large enough relative to the genus of $C$. (Extra credit: good bound on the degree needed.)

[1] A quadric is a degree $2$ hypersurface, i.e., the zero set in $\mathbf{P}^ r$ of a degree $2$ homogeneous polynomial.

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