Another reference request about dualizing sheaves for nodal surfaces

Question

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.

Answer 1

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].

Answer 2

Your question is local on $\Sigma$, so you can consider the case where $\Sigma = \mathbb{C}^2/G$ where $G\subset SL_2$ is a finite subgroup. Then $\omega_\Sigma$ is the trivial line bundle (the trivialization of $\omega_{\mathbb{C}^2}$ given by $dz_1 dz_2$ descends to $\omega_\Sigma$ since it is acted on invariantly by $G$) so it suffices to show that $K_S$ is trivial for $S\to \mathbb{C}^2/G$ the minimal resolution. This triviality follows from the adjunction formula once you know that the exceptional divisor is a configuration of $\mathbb{P}^1$s whose normal bundles are $\mathcal{O}(-2)$. This fact can be found in a bunch of places, the first one that comes to mind is Shafarevich's book.

One slick way to simultaneously construct the resolution and to show it has trivial canonical bundle is to let $S = G\operatorname{-Hilb}(\mathbb{C}^2)$, that is the $G$ fixed points in the Hilbert scheme of $|G|$ points on $\mathbb{C}^2$. Note that $G\operatorname{-Hilb}(\mathbb{C}^2)$ admits a birational morphism to $\mathbb{C}^2/G$ (sending a $G$-invariant configuration of $|G|$ points to the corresponding orbit). Now the Hilbert scheme of points is non-singular and has trivial canonical bundle, and so the fixed point set of a $G$ action is also non-singular and has trivial canonical bundle, hence $K_S$ is trivial. This is explained in Nakajima's book or Dogachev's book (lecture 6)

http://www.math.lsa.umich.edu/~idolga/McKaybook.pdf