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On the Irreducibility of Commuting Varieties of Nilpotent Matrices

R. Basili
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Abstract

Given an nxn nilpotent matrix over an algebraically closed field K, we prove some properties of the set of all the nxn nilpotent matrices over K which commute with it. Then we give a proof of the irreducibility of the variety of all the pairs (A,B) of nxn nilpotent matrices over K if either char K = 0 or char K isn't less than n/2. We get as a consequence a proof of the irreducibility of the local Hilbert scheme of n points of a smooth algebraic surface over K with the previous condition on char K.

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R. Basili. 2003-01-20. On the Irreducibility of Commuting Varieties of Nilpotent Matrices. https://arxiv.org/abs/math/0301215

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