  Let M_d(k) denote the space of dxd-matrices with coefficients in an
algebraically closed field k. Let X be an orbit closure in the product
[M_d(k)]^t equipped with the action of the general linear group GL_d(k) by
simultaneous conjugation. We show that X is regular at any its point y such
that the orbit of y has codimension one in X. The proof uses mainly the
representation theory of associative algebras.
