Classification of 3-dimensional isolated rational hypersurface singularities with \(\mathbb{C}^*\)-action
Rational double points of surfaces have been classified by \textit{M. Artin} [Am. J. Math. 88, 129--136 (1966; Zbl 0142.18602)] about forty years ago. He identified their equations in a list which is usually denoted ADE. They are known to be the absolutely isolated double points of surfaces which appear (up to quadratic suspension) on hypersurfaces in all dimensions. But there is another way to generalize this notion to higher dimensions: A rational singularity is defined by the condition, that the higher direct images of the structure sheaf on a resolution vanish. Compared to surfaces, the classification of 3-dimensional isolated rational hypersurface singularities with \(\mathbb C^*\)-action gives many additional cases, classified by the authors in the article under review. Let \(h\) be a complex weighted homogeneous polynomial in 4 variables, then its zero-set \(V\) can be deformed into one of 19 (known) classes of weighted homogeneous singularities while keeping the differential structure of the link \(K_V:=V\cap S^7\) constant. The authors show that those deformations actually preserve weights and the embedded topological type. Rationality is equivalent with vanishing of the geometric genus \(p_g\) of the singularity. Using a theorem of \textit{M. Merle} and \textit{B. Teissier} [Semin. sur les singularites des surfaces, Cent. Math. Ec. Polytech., Palaiseau 1976--77, Lect. Notes Math. 777, 229--245 (1980; Zbl 0461.14009)] the number \(p_g\) can be calculated from the weights of the weighted homogeneous polynomial. The authors give the list of equations of rational 3-dimensional isolated hypersurface singularities with \(\mathbb C^*\)-action for all cases, where conditions on the exponents are expressed by a linear form. Evaluating these conditions leads to a list of about 17 pages, which can be found in the extended online-version of the paper [\texttt{math.AG/0303302}]. This valuable result is given without detailed proof, some parts are obtained using a computer algebra system. Reviewer's remark: Of course, when the computer wasn't part of a mathematicians day to day life it was usual as well to omit lengthy and complicated calculations. But it may be questioned today, whether it wasn't appropriate to include more details at least in the electronic version, since otherwise in a reasonable time it is nearly impossible to check such a list for correctness. Note that there was an alternative way the authors could obtain the rational cases in the 19 general classes, using a result of \textit{H. Flenner} [Arch. Math. 36, 35--44 (1981; Zbl 0454.14001)]: Let \((\omega _1, \dots ,\omega _4)\) be the normalized weights and \(s= 4-2\cdot \sum \omega _i\) Saito's invariant. Then rationality is equivalent to \(s<2\).
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