# 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? – Mohan Apr 25 '17 at 22:24
• Yes. What are you implying? – Monstrous Moonshine Apr 25 '17 at 22:25
• They are typical cases when the canonical bundle is not very ample. – Mohan Apr 26 '17 at 2:17
• Oh, ok. Is there any sufficient condition for the canonical divisor being very ample? – Monstrous Moonshine Apr 26 '17 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. – Bilateral Jul 9 '18 at 22:15