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Today i was talking with my advisor and she told me the following fact:

Let $S$ be a singular surface in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$. Writing $\omega_\Sigma$ for the dualizing sheaf and $H_S$ for a hyperplane section, then we have $$ \omega_{S} = (d-4)H_S. $$ I.e. there is an adjunction type formula with the canonical sheaf replaced by the dualizing sheaf.

This is supposed to hold for any variety whose singularities are Gorenstein.

I would like to include this in my thesis but i need a reference. Does anyone know a good one? Thanks.

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    $\begingroup$ This may be a silly question but... have you tried asking your advisor? $\endgroup$ – Mariano Suárez-Álvarez Mar 27 '13 at 13:52
  • $\begingroup$ I agree with Mariano. But let me give you some hitns. At some level this also all is in Residues and Duality by Hartshorne. The above sort of formula (say for a Cartier divisor) comes from for example Hom'ing $$0 \to O(-H) \to O \to O_H \to 0$$ into the dualizing sheaf/complex of the ambient space, and then taking cohomology. $\endgroup$ – Karl Schwede Mar 27 '13 at 14:01
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    $\begingroup$ @Mariano, yes i did. $\endgroup$ – Joachim Mar 27 '13 at 14:05
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One reference is [Hartshorne, Algebraic Geometry, Theorem 7.11 p. 245].

For the reader's convenience, I will restate it here.

Theorem Let $X$ be a closed subscheme of $P=\mathbb{P}^n$ which is a local complete intersection of codimension $r$. Let $\mathscr{I}$ be the ideal sheaf of $X$. Then $$\omega_X^{\circ} \cong \omega_P \otimes \wedge^r (\mathscr{I}/\mathscr{I}^2)^{\vee},$$ where $\omega_X^{\circ}$ denotes the dualizing sheaf.

In your case, since $X=S$ is a codimension $1$ subvariety of $\mathbb{P}^3$, one has $$\omega_P=\mathscr{O}_P(-4), \quad (\mathscr{I}/\mathscr{I}^2)^{\vee}=\mathscr{O}_S(d)=dH_S$$ so the claim follows.

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  • $\begingroup$ Ok, i feel stupid now, thanks a lot for your clear answer! $\endgroup$ – Joachim Mar 27 '13 at 14:06

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