# Canonical map for algebraic curve is nondegenerate

algebraic-curvesalgebraic-geometry

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)?

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.