Simple singularities in positive characteristic
Simple singularities (also called ADE because the different types \(A_ k\), \(D_ k\) and \(E_ 6\), \(E_ 7\), \(E_ 8\) in which they are classified) appear in a natural way in problems of classification of singularities from different points of view. In dimension 1, simple singularities have been characterized in terms of their resolution procedure by \textit{W. P. Barth}, \textit{C. A. M. Peters} and \textit{A. J. H. M. Van de Ven} [``Compact complex surfaces'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Band 4 (1984; Zbl 0718.14023)] in characteristic 0 and in any characteristic by \textit{K. Kiyek} and \textit{G. Steinke} [Arch. Math. 45, 565-573 (1985; Zbl 0553.14012)] who obtained the normal forms of them. In dimension 2 simple singularities are just the rational double points [see \textit{M. Artin}'s paper in Complex Anal. Algebr. Geom., Collect. Papers dedic. K. Kodaira, 11-22 (1977; Zbl 0358.14008)]. On the other hand in characteristic 0 and arbitrary dimension \textit{V. I. Arnol'd} showed [cf. Funct. Anal. 6(1972), 254-272 (1973); translation from Funkts. Anal. Prilozh. 6, No.4, 3-25 (1972; Zbl 0278.57011)] that they are exactly the hypersurface singularities of finite deformation type (i.e. singularities from which one can obtain, by deformation, only a finite number of nonequivalent singularities). According to previous works of \textit{H. Knörrer} [Invent. Math. 88, 153- 164 (1987; Zbl 0617.14033)] and \textit{R.-O. Buchweitz}, \textit{G.-M. Greuel} and \textit{F.-O. Schreyer} [ibid. 165-182 (1987; Zbl 0617.14034)] it is known that the simple singularities are the hypersurface singularities with finite Cohen-Macaulay type (i.e. singularities for which there exists only a finite number of non isomorphic indecomposable maximal Cohen-Macaulay modules over their local ring). However, the relationship with the point of view of Arnol'd (using deformation theory) remains essentially unknown in arbitrary characteristic. This point constitutes the main result in the paper, more precisely, if \(f\in kx_ 1,...,x_ n\), k being an algebraically closed field of arbitrary characteristic, the authors define f to be simple if and only if f is contact-equivalent with one of the normal forms given in dimension 1 by Kiyek and Steinke, in dimension 2 by Artin and for dimension greater than two by double suspension of curves or surfaces. Then, the main theorem asserts that the following statements are equivalent: (a) f is simple; (b) f is of finite deformation type; and (c) f is of finite Cohen-Macaulay type. - In particular a complete list of the normal forms for simple singularities is given, and, although the calculations in order to obtain the normal form of a given hypersurface are not completely described, a useful list of the main subcases is included. Also the adjacencies between the different types of simple singularities are completely given except for the case of surface singularities in characteristic 2 in which only partial information appears [see also \textit{F. Knop}, Invent. Math. 90, 579-604 (1987; Zbl 0648.14002)]. The characterizations of the singularities \(A_{\infty}, D_{\infty}\) are also obtained by the same methods. The differences to the well known case of characteristic 0 are adequately explained and some useful examples are given in order to make clear the exceptions appearing only in positive characteristic.
hypersurface singularities of finite deformation type
hypersurface singularities with finite Cohen-Macaulay type
scientific article; zbMATH DE number 4177241