The normality of closures of conjugacy classes of matrices (Q1177119)

Let \(K\) be an algebraically closed field, \(n\) a positive integer, \(x\) an \(n \times n\) matrix with entries in \(K\), let \(C(x)\) be the orbit of \(x\) under the adjoint action of \(\text{GL} (n,K)\) and let \(\overline {C(x)}\) be the Zariski closure of \(C(x)\). \textit{H. Kraft} and \textit{C. Procesi} [ibid. 53, 227-247 (1979; Zbl 0434.14026)], have proved that if \(K\) has characteristic 0 then \(\overline {C(x)}\) is a normal, Cohen-Macaulay variety with rational singularities. The main purpose of this paper is to prove, in arbitrary characteristic, that \(\overline {C(x)}\) is normal.