Recognition of simple singularities in positive characteristic (Q1202213)

The classification by \textit{G. M. Greuel} and \textit{H. Kröning} in Math. Z. 203, No. 2, 339-354 (1990; Zbl 0715.14001) yields a list of equations of type \(A\), \(D\) and \(E\), which is reproduced in an annex to this paper. The author shows how to recognise these singularities from their weights. Let \(X=\text{Spec} R\), with \(R\) a complete Cohen-Macaulay local \(k\)- algebra with algebraically closed residue field \(k\) of dimension \(n\geq 1\). Then \(X\) is of type \(A\), \(D\) or \(E\), if and only if \(X\) is semi- quasihomogeneous of some weight \(w\) \((=(w_ 1,\dots,w_ n))\) such that \(w_ 0+\cdots+w_ n<n/2\). For \(n>1\) the proof uses the characterisation of the \(ADE\) singularities as absolutely isolated double points, which is here established in positive characteristic for \(n>2\).