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.

Residues and Dualityby 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. $\endgroup$