My advisor told me the following:


Let $\Sigma$ be a singular surface over $\mathbb{C}$ whose singularities are all ordinary quadratic, or more generally Duval singularities. Let $\epsilon: S \rightarrow \Sigma$ be the desingularization. Then, writing $\omega\_\Sigma$ for the dualizing sheaf of $\Sigma$ and $K\_S$ for the canonical sheaf on $S$ we have
$$
\epsilon^\*\omega\_\Sigma = K\_S.
$$
A reference would be Duval's original paper, but this is quite old. Does anyone know a better one?


We tried Reid's chapters on algebraic surfaces and some other obvious ones but did not find it there.



Another reference request about dualizing sheaves for nodal surfaces
I assume that $\epsilon \colon S \to \Sigma$ is the minimal resolution of the singularities. 
Then $$\omega\_S=\epsilon^\* \omega\_{\Sigma}+\sum a\_iE\_i,$$  

where the $E\_i$ are the exceptional divisors and $a\_i \leq 0$. 


Then $\epsilon^\* \omega\_{\Sigma}=\omega\_S$ if and only if the surface $\Sigma$ has *canonical singularities*. So one must prove that canonical singularities in dimension $2$ coincide with Du Val singularities, i.e. Rational Double Points. 


This is well-known and it is explained in several places. Good references are [Matsuki, *Introduction to the Mori Program*, Theorem 4-6-7 p. 197] or [Kollar-Mori, *Birational Geometry of Algebraic Varieties*, Theorem 4.20 p. 122]. 


