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Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. Appendix 0: A good generating set in the case of \(E 8\). Appendix 1: Standard coordinates of \(E 6\). Appendix 2: Standard coordinates of \(E 7\). (English) Zbl 0788.14036
For every small resolution \(\pi:Y\to X\) of a Gorenstein threefold singularity with irreducible exceptional curve \(C\), there exists a second small resolution \(\pi^ +:Y^ +\to X\); the birational map \(Y \to Y^ +\) is a simple flop. The general hyperplane section of \(X\) is a rational double point. Every small resolution \(\pi:Y\to X\) can be constructed from a simultaneous resolution of a one-parameter deformation of a rational double point; this singularity is in general not isomorphic to the general hyperplane section. This paper shows that only \(A_ 1\), \(D_ 4\), \(E_ 6\), \(E_ 7\) and \(E_ 8\) occur as general hyperplane section; the type is determined by the multiplicity of the maximal ideal along the exceptional curve.
To prove their result, the authors use an explicit description of the simultaneous resolution of the versal deformation of the \(A\)-\(D\)-\(E\)- singularities. They compute and manipulate with the invariants of the corresponding Weyl groups. For \(E_ 8\) the resulting formula’s are not given, because they are too long, but the algorithms are described (the computations were done in MAPLE and REDUCE). For \(E_ 7\) some unusual coefficients are used, to make the results comparable to those of C. C. Bramble [Am. J. Math. 40, 351-365 (1919)].

14J30 \(3\)-folds
14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
32S25 Complex surface and hypersurface singularities