Regularity in codimension one of orbit closures in module varieties

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