A geometric reference for (affine) Gorenstein varieties and singularities - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:14:07Z http://mathoverflow.net/feeds/question/57505 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57505/a-geometric-reference-for-affine-gorenstein-varieties-and-singularities A geometric reference for (affine) Gorenstein varieties and singularities aglearner 2011-03-05T21:48:03Z 2011-03-07T00:43:21Z <p>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?</p> <p>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. </p> http://mathoverflow.net/questions/57505/a-geometric-reference-for-affine-gorenstein-varieties-and-singularities/57561#57561 Answer by Karl Schwede for A geometric reference for (affine) Gorenstein varieties and singularities Karl Schwede 2011-03-06T12:23:56Z 2011-03-06T18:19:55Z <p>Hi, 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.</p> <p>Another algebraic place to read about Gorenstein singularities (besides Eisenbud's book) include Bruns and Herzog's <em>Cohen-Macaulay rings</em>.</p> <p>There is also a question you should ask yourself about Gorenstein singularities. Which of the following properties of Gorenstein singularities do you want:</p> <ol> <li>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 <em>Algebraic Geometry</em>). </li> <li>The fact that on a Gorenstein variety, the <em>canonical</em> Weil divisor $K_X$ is actually a Cartier divisor.</li> </ol> <p>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.</p> <p>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).</p> <p>Suppose that X is a projective variety with an ample line bundle $\mathcal{L}$. Then the section ring:</p> <p>$$\oplus_{n \geq 0} H^0(X, \mathcal{L}^{\otimes n})$$</p> <p>is Gorenstein if and only if the following two conditions hold.</p> <ol> <li><p>$H^i(X, \mathcal{L}^{\otimes n}) = 0$ for all $i > 0$ and all $n \geq 0$. This is just condition 1. above.</p></li> <li><p>$\mathcal{O}_X(K_X)$ is isomorphic to $\mathcal{L}^n$ for some integer $n$. This is condition 2. above.</p></li> </ol> <p><strong>EDIT:</strong> If you have explicit equations, you can often use <a href="http://www.math.uiuc.edu/Macaulay2/" rel="nofollow">Macaulay2</a> to check whether the ring is Gorenstein. Let me know if this would be useful to you.</p> http://mathoverflow.net/questions/57505/a-geometric-reference-for-affine-gorenstein-varieties-and-singularities/57596#57596 Answer by Hailong Dao for A geometric reference for (affine) Gorenstein varieties and singularities Hailong Dao 2011-03-06T20:17:35Z 2011-03-07T00:43:21Z <p>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 <a href="http://www.math.ucdavis.edu/~osserman/classes/256A/notes/cheat-props.pdf" rel="nofollow">here</a>. So it is unlikely you can find such reference. </p> <p>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 <em>not</em> 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 <a href="http://mathoverflow.net/questions/55526/example-of-a-variety-with-k-x-mathbb-q-cartier-but-not-cartier/55531#55531" rel="nofollow">recent comment</a> that most people don't even realize that it could be an issue. </p> <p>But without Cohen-Macaulayness, the canonical sheaf would not be truly <em>dualizing</em> (see the comments <a href="http://mathoverflow.net/questions/48145/dualizing-sheaf/48146#48146" rel="nofollow">here</a>). This is perhaps where the real power of the property lies. </p> <p>Finally, I would like to recommend this <a href="http://front.math.ucdavis.edu/0209.5199" rel="nofollow">survey on Gorenstein rings</a>. You can pick up a lot about them from there, including the very interesting history. To quote from the Introduction:</p> <blockquote> <p>As we shall see, they could perhaps more justiﬁably be called Bass rings, or Grothendieck rings, or Rosenlicht rings, or Serre rings. </p> </blockquote> <p>Enjoy! </p>