# When is the canonical sheaf of a curve very ample?

Let $X/k$ be a smooth projective curve over an algebraically closed field $k$ of genus $g$, then when is it that the canonical sheaf $\omega_X$ is very ample, i.e. $\omega_X = i^*\mathcal{O}_{\mathbb{P}^n}(1)$ for some closed immersion $i:X\hookrightarrow\mathbb{P}^n$? My intuition is that this is true for any $g\gg0$, and probably for something like $g\ge 2$ or $g\ge 3$, but I don't immediately see how to prove this. Is Riemann-Roch the correct approach?

• Have you heard of Hyperelliptic curves? Commented Apr 25, 2017 at 22:24
• Yes. What are you implying? Commented Apr 25, 2017 at 22:25
• They are typical cases when the canonical bundle is not very ample. Commented Apr 26, 2017 at 2:17
• Oh, ok. Is there any sufficient condition for the canonical divisor being very ample? Commented Apr 26, 2017 at 2:18
• I don't understand why so much hate in the comments. @Sasha, basically every question can be answered by looking into the appropriate book, so your comment es irrelevant and inappropriate. Plus, this is a site for comments of all levels. Having a constructive answer would be great. Commented Jul 9, 2018 at 22:15

Proposition 5.2 Let $$X$$ be a curve of genus $$g \geq 2$$. Then $$|K|$$ is very ample if and only if $$X$$ is not hyperelliptic.
As by Exercise IV 1.7. there are hyperelliptic curves of any genus $$g$$, the very-ampleness of $$K$$ is independent of the genus (save for the cases $$g = 0$$ and $$g = 1$$, in which $$K$$ has degree $$\leq 0$$, and hence can never be (very) ample).