references for singularities: does quotient singularities imply gorenstein? Is there a good place where to learn about singularities of algebraic varieties?
OK, this question is terribly vague, so I'll ask what I'm currently interested in: if X is a smooth variety and G is a finite group, then is $X/G$ Gorenstein? If true, what would a reference be?
 A: Here is a related question on mathoverflow.
It sounds like the relevant part of 'Gorenstein' here is that the canonical divisor is Cartier.  Or, in other words, the canonical sheaf is invertible.  This is not necessarily true for quotients of smooth varieties.
Here is a simple example of an isolated quotient singularity which is non-Gorenstein:  Let $X \cong \mathbb{C}^3 = \mathrm{Spec}\,\mathbb{C}[z_1,z_2,z_3]$ and let $G \cong \mathbb{Z}_2$ act on $X$ as $(z_1, z_2, z_3) \to (-z_1, -z_2,-z_3)$.  Then $X/G = \mathrm{Spec}\,\mathbb{C}[z_1^2,z_2^2,z_3^2,z_1 z_2, z_1 z_3, z_2 z_3]$, and this is not Gorenstein.  There are (at least) a couple of ways to see this:


*

*If you know toric geometry, it is a simple exercise to show that $K_{X/G}$ is not Cartier.

*Roughly, the stalk of $\omega_X$ at the origin is generated by $dz_1\wedge dz_2\wedge dz_3$, and this is not preserved by $G$, so $\omega_{X/G\setminus\{0\}}$ cannot be extended to a line bundle at the origin.  In more detail, $\omega_{X/G}$ has three independent sections in a neighbourhood of the origin:
$z_1dz_1\wedge dz_2\wedge dz_3~,~ z_2dz_1\wedge dz_2\wedge dz_3~,~ z_3dz_1\wedge dz_2\wedge dz_3$.

A: The standard textbook references for learning singularities and their resolution are:


*

*Greuel/Lossen/Shustin - Introduction to Singularities And Deformations, Springer 2007.

*Kollár, J. - Lectures on Resolution of Singularities, Princeton University Press 2007.

*Cutkosky, S. D. - Resolution of Singularities, AMS Graduate Studies in Mathematics 2004.

*Kollár/Kovács - Singularities of the Minimal Model Program, Springer 2013.

*Hauser et al. - Resolution of Singularities, a Research Textbook, Birkhauser 2000.


Specially, Kollár's and Cutkosky's are great books giving many examples and proving Hironaka's famous resolution theorem. Maybe you could find there the kind of topics you are interested in. For an introduction to singularities and a deeply conceptual overview towards such a theorem, there is the fantastic freely available article:


*

*Hauser, H. - The Hironaka theorem on resolution of singularities (Or: a proof we always wanted to understand), Bulletin of the AMS vol. 40 no.3, p.323-403, 2003.

