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I am interested in the question of the classification of (low rank) Gorenstein rings over $\mathbb{C}$. The socle of a local algebra is the annihilator of its maximal ideal. A commutative local ring is called Gorenstein iff its socle is 1-dimensional. Here, "rank" means the dimension of the algebra as a vector space.

A classification of low rank (<7) algebra can be found in http://math.mit.edu/~poonen/papers/dimension6.pdf and so do the Gorenstein rings. But I didn't find further results. Is there any reference?

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This paper might possibly be relevant:

http://www.ams.org/mathscinet-getitem?mr=2922602

It's not quite exactly what you are asking for. Instead of Gorenstein algebras of low rank, they are considering short algebras, i.e., ones with small socle degree. But hopefully it is related and helpful for you.

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