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?
Asked
Active
Viewed 891 times
4
-
1Have you heard of Hyperelliptic curves? – Mohan Apr 25 '17 at 22:24
-
Yes. What are you implying? – Dominic Wynter Apr 25 '17 at 22:25
-
1They are typical cases when the canonical bundle is not very ample. – Mohan Apr 26 '17 at 02:17
-
Oh, ok. Is there any sufficient condition for the canonical divisor being very ample? – Dominic Wynter Apr 26 '17 at 02:18
-
8I 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 09 '18 at 22:15
1 Answers
3
Hartshorne, Proposition IV.5.2 has an answer, and I guess this is what Mohan hinted at in the comments:
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).
red_trumpet
- 8,515