Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\Omega_X$ be the canonical bundle. Then $\Omega_X$ is a line bundle of degree $2g-2$, where $g$ is the genus of $X$, and for $g>1$ it is base point free and determines a map $X\to\Bbb P^{g-1}$ called the canonical map. When $X$ is hyperelliptic, the image is a rational normal curve and the map is the 2-to-1 cover $X\to\Bbb P^1$ followed by the $(g-1)$-uple embedding; when $X$ is not hyperelliptic, the map is a closed immersion.

In the first case, it follows from the properties of rational normal curves that the image of the canonical map is nondegenerate (not contained in a hyperplane). I feel like this should also hold in the second case, so that we can say the image of the canonical map is always nondegenerate. But I don't know how to justify this when the canonical map is a closed immersion (and I feel a little silly for not immediately knowing the answer). Can you help me prove this (or provide a counterexample in case it's not true)?


1 Answer 1


The image of canonical map is not contained in any hyperplane follows by the definition: If it were, then the hyperplane corresponds to a 1-form $\omega\in H^0(\Omega)$ which vanish on $X$ everywhere.

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    $\begingroup$ And this doesn't exist because $\Omega$ is a line bundle and $X$ is reduced, so locally such a thing is an element of $\mathcal{O}_X(U)$ which vanishes everywhere but is nonzero. Okay, I think I've got it now - thanks! $\endgroup$ Commented Jan 1, 2022 at 3:46

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