A geometric reference for (affine) Gorenstein varieties and singularities

Question

I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a person whose knowlage of commutative algebra is minor). What standard things one could try to do to check if a given scheme is Gorenstein?

If the field of definition is $\mathbb C$ - it is even better. I know one source - Eisenbud "commutative algebra" but find it a bit hard.

Answer 1

I don't think there is an easy way to do this in general. Gorenstein is a fairly homological / commutative algebraic condition. However, the condition that $K_X$, the canonical divisor, is a Cartier divisor is quite close to the Gorenstein condition and for some purposes, it is just as good.

Another algebraic place to read about Gorenstein singularities (besides Eisenbud's book) include Bruns and Herzog's Cohen-Macaulay rings.

There is also a question you should ask yourself about Gorenstein singularities. Which of the following properties of Gorenstein singularities do you want:

1. The fact that Gorenstein singularities are Cohen-Macaulay (and so have well-behaved Serre-duality without the need for fancy homological machinary and derived categories, see the Serre duality section in Hartshorne's Algebraic Geometry).
2. The fact that on a Gorenstein variety, the canonical Weil divisor $K_X$ is actually a Cartier divisor.

In fact, a singularity being Gorenstein is equivalent to both conditions 1. and 2. I also think that 1. + 2. is how most geometers think about the Gorenstein condition. Commutative algebraists tend to have a different perspective.

I should also point out perhaps one other large class of rings where you can easily detect whether or not it is Gorenstein (besides the already-mentioned complete intersections).

Suppose that X is a projective variety with an ample line bundle $\mathcal{L}$. Then the section ring:

$$ \oplus_{n \geq 0} H^0(X, \mathcal{L}^{\otimes n})$$

is Gorenstein if and only if the following two conditions hold.

1. $H^i(X, \mathcal{L}^{\otimes n}) = 0$ for all $i > 0$ and all $n \geq 0$. This is just condition 1. above.

2. $\mathcal{O}_X(K_X)$ is isomorphic to $\mathcal{L}^n$ for some integer $n$. This is condition 2. above.

EDIT: If you have explicit equations, you can often use Macaulay2 to check whether the ring is Gorenstein. Let me know if this would be useful to you.

Answer 2

Sorry, tough luck, but most first (and second) algebraic geometry courses don't even touch Cohen-Macaulay rings, let alone Gorenstein. Look, for example, at Definition 4.2 here. So it is unlikely you can find such reference.

Here is an explanation why you need both Cohen-Macaulayness and the fact that the canonical divisor is Cartier, as mentioned in Karl's answer. The trouble is that there are UFDs (so all divisors are Cartier) which are not Cohen-Macaulay (for instance, the invariant ring of $\mathbb Z_4$ acting by cyclically permuting the variables on the polynomial ring in four variables over a field of char 2). Such examples are not very well-known, I remember Sándor Kovács pointed out in a recent comment that most people don't even realize that it could be an issue.

But without Cohen-Macaulayness, the canonical sheaf would not be truly dualizing (see the comments here). This is perhaps where the real power of the property lies.

Finally, I would like to recommend the survey on Gorenstein rings by Craig Huneke: Hyman Bass and Ubiquity: Gorenstein Rings. You can pick up a lot about them from there, including the very interesting history. To quote from the Introduction:

As we shall see, they could perhaps more justifiably be called Bass rings, or Grothendieck rings, or Rosenlicht rings, or Serre rings.

Enjoy!