# Non-Gorenstein Curves

I am interested in (as explicit as possible) descriptions of non-Gorenstein integral projective curves. Most of the literature on singular curves appears to be focused around the Gorenstein case, with a notable exception being the paper of Kleiman, The Canonical Model of a Singular Curve, and a handful of others. Any other references would be appreciated. Specifically I am interested in knowing what has been understood about their derived categories of coherent sheaves (or the subcategory of perfect complexes), so I would very interested in any developments along those lines.

## 1 Answer

I don't think there is a reasonably short explicit description of non-Gorenstein curves. They could be described as lacking some of those conditions that define Gorenstein curves.

In any case, here is a simple example: Let $$C$$ be the union of three coordinate axis in $$\mathbb A^3$$. Then $$C$$ is not Gorenstein. It is probably a good exercise for you to check that this is true.

What I find fascinating about this is that the union of three lines contained in $$\mathbb A^2$$ is a Gorenstein curve, so this shows that being Gorenstein or not could depend on very subtle details.

Of course, $$C$$ can be projected to a plane and if the projection is general, then it will be one-to-one on $$C$$ and an isomorphism on each component to their respective images, however, it is not an isomorphism on $$C$$. This follows from the fact that the image of $$C$$ is Gorenstein, or more directly from the fact that the Zariski tangent space of $$C$$ at its singular point is $$3$$-dimensional while that of its image is $$2$$-dimensional, so they can't be isomorphic.

(In case you are worried that this example is reducible, then think about how to get this singularity on an irreducible curve).

• Thanks for your answer. I would say that an integral curve in $\mathbb{A}^3$ with a singularity at the origin analytically isomorphic to $C$ above would be non-Gorenstein, yes? If no one is able to say anything about the derived categories of such curves (or provide any other references) in a bit, I will accept your answer. – DKS Apr 30 at 15:39
• Yes, that's right. – Sándor Kovács Apr 30 at 19:12