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.



Reference for fact about dualizing sheaf of singular varieties
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.


