On the Irreducibility of Commuting Varieties of Nilpotent Matrices

Details On the Irreducibility of Commuting Varieties of Nilpotent Matrices On the Irreducibility of Commuting Varieties of Nilpotent Matrices

2003-01-20

arxiv.org/abs/math/0301215v1

Mathematics

Algebraic Geometry

LaTex, 27 pages, to appear in J. Algebra

Scientific paper

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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