# Strange case of Serre's duality

$\newcommand{\O}{\mathcal{O}}$ Let $X$ be a smooth projective curve and $D$ and effective divisor on it. The normal bundle of $D$ is defined as $$\O_D(D)\; = \; \O_x(D)\;\otimes_{\O_X}\, \O_D$$ where $\O_D$ is just the restriction of the stricture sheaf of $X$ to the support of $D$. In the literature I found the claim that, as a special case of Serre's duality, the dual space to $H^0(X, \O_D(D))$ is $$H^0(X, \O_D(D))^* \cong H^0(X,K\otimes\O_D),$$ where $K$ is the canonical sheaf.

Now, I'm not an expert of Serre's duality, but I was expecting the dual of that space to be something like $$H^0(X,(K-D)\otimes\O_D).$$ Could you please explain the reason why the above is the right answer and mine is the wrong one?

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The adjunction formula says that the dualizing sheaf $\omega_D$ on $D$ is $K\otimes O_D(D)$. For any invertible sheaf $L$ on $D$, we have $H^0(D, L)^*=H^0(D, \omega_D\otimes L^{-1})$. Now apply this to $L=O_D(D)$. – Cantlog Oct 20 '13 at 9:03
Thanks! I found it in Hartshorne II - 8.20 (page 182) – Abramo Oct 20 '13 at 9:12
Ah, then please write and accept your answer. – Cantlog Oct 20 '13 at 9:13
Dear Abramo and @Cantlog, beware that Hartshorne's result only works in the special case that $D$ is smooth. – Georges Elencwajg Oct 20 '13 at 10:08
Dear Georges, @GeorgesElencwajg: too bad. The adjunction formula holds in general for any effective Cartier divisor. But I can't find it in Fulton's "Intersection Theory". Maybe in SGA 6 ? – Cantlog Oct 20 '13 at 17:10

From Proposition 8.20 of Hartshorne (chapter II, page 182), we have a formula for the canonical sheaf $\omega_D$ of $D$: $$\omega_D = \omega_X \otimes \O_D(D).$$ So applying Serre's duality for $D$ we find $$H^0(\O_D(D))^* \cong H^0(\omega_D \otimes \O_D(D)^{-1}) \cong H^0(\omega_X \otimes\O_D(D)\otimes \O_D(D)^{-1}) \cong H^0(\omega_X \otimes \O_D).$$
Remark: we are just using the hypothesis of Serre duality from Serre's book, so that $X$ is a smooth projective curve over an algebraically closed field.