Determinantal singularities (Q6608480)

This chapter of the book is devoted to the study of determinantal singularities of analytic spaces with an emphasis on deformation theory, elementary methods of classification and related computational methods with applications. It consists of five parts, a list of 114 references on relevant articles and 18 classification tables of simple singularities of various kinds and types, most of which were obtained by the authors in a series of earlier works (see, e.g., [\textit{A. Frühbis-Krüger}, Commun. Algebra 27, No. 8, 3993--4013 (1999; Zbl 0963.14011); \textit{A. Frühbis-Krüger} and \textit{A. Neumer}, Commun. Algebra 38, No. 2, 454--495 (2010; Zbl 1193.32015); \textit{M. Zach}, Topology of isolated determinantal singularities. (PhD Thesis) (2017)]).\N\NThe authors begin with preliminary materials and a collection of facts known to them concerning singularities of matrices and determinantal varieties. Among other things they discuss the basic properties of determinantal ideals and their free resolutions, deformations of determinantal singularities, the concepts of unfoldings and equivalence of matrices, finite determinacy, versality, etc. Then the theory of essentially isolated determinantal singularities is considered in detail including the Tyurina transformation for such singularities and their deformations, possible generalizations, modifications and so on (see [\textit{A. Frühbis-Krüger}, Topology Appl. 234, 375--396 (2018; Zbl 1405.32038)]). In the next two parts, they discuss the classification problem of simple complete intersections, Cohen-Macaulay singularities of codimension 2, Gorenstein singularities of codimension 3, rational surface singularities, singularities of square matrices, and some other types. The last part is concerned to analysis of the topology of essential smoothings of essentially isolated determinantal singularities, explicit constructions of smoothings, some special properties of determinantal hypersurfaces, isolated Cohen-Macaulay singularities of codimension 2 and some other particular cases and applications, including properties of hypersurfaces with nonisolated singularities, bouquet decompositions for essential smoothings (see [\textit{M. Zach}, J. Singul. 22, 190--204 (2020; Zbl 1451.14007)]), formulas for the vanishing Euler characteristic and complex links of essential smoothings, etc.\N\NIt should be especially noted that the exposition is illustrated with 12 excellent colour pictures with interesting comments and useful remarks.\N\NFor the entire collection see [Zbl 1545.32001].