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This question is motivated by the question Subalgebras of matrices and its answer by Mariano. We consider $X_{n,d}$, the variety of $d$-dimensional subalgebras not necessarily with 1 (with 1 makes sense too) in $M_n({\mathbb C})$. What are its irreducible components?

If $d=1$, it is a cute little exercise on Jordan normal forms (please, donate 5 bucks of my royalties to Oxfam if you are to use it on your students). A full answer for $d=2$ will be given full marks here :-))

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