Nonlinear sections of nonisolated complete intersections. (Q2754600)

This paper is devoted to nonisolated singularities (NIS). Hamm (1971) extended the study of isolated hypersurfaces singularities (IHS), initiated by Milnor (1968) for IHS, and to isolated complete intersection singularities (ICIS). The theory of Hamm and others deals with the notion of Milnor fibration as a principal tool and various other techniques of topological and local geometrical nature. \textit{E. J. N. Looijenga} had presented a complete exposition of all results obtained [Isolated singular points on complete intersections. (Lecture Notes Cambridge Univ. Press 1984; Zbl 0552.14002)]. NEWLINENEWLINEAccording to the author of the paper under review, two fundamental obstacles appear in the case of NIS: first, the Milnor fibration does not possess connectivity properties like in the case of ICIS; second, the computation of the relative singular Milnor numbers of a divisor of an ICIS as a length of a determinable module does not hold in the case of nonisolated complete intersection singularities (NICIS). NEWLINENEWLINEThe main purpose of the author is to describe and analyze the intrinsic nature of the results obtained in the last years for nonisolated singularities (NIS). An alternative approach, originating in algebraic geometry, is developed in this paper. It applies to large classes of NIS arising as nonlinear sections of fixed ``model nonisolated singularities''. Thom-Mather type of the group of equivalence acting on the sections and their unfoldings is used in the study of singularities of such sections. A singular analogue of the Milnor fibration for nonlinear sections of NIS is obtained. It retains the same connectivity properties as for Milnor fibers of ICIS. The notion of freeness for divisors [\textit{K. Saito}, J. Fac. Sci. Univ. Tokyo Sect. Math. 27, 265--291 (1980; Zbl 0496.32007)] provides the algebraic condition needed for the computation of singular Milnor numbers in the above mentioned sense. NEWLINENEWLINEDe Rham cohomology methods are described in connection with the study of the topology of nonlinear sections. More concretely: the cohomology of the complement of a free divisor, certain local cohomologies, the singular Milnor fiber and Gauss-Manin connection for the singular Milnor fibration. Some results on freeness of discriminants are summarized: Cohen-Macaulay reduction, Morse-type singularities. The structure of modules and especially determinantal modules are considered. NEWLINENEWLINEFinally, one can say that the author's philosophy is well expressed as follows: ``We raise several natural questions regarding the intrinsic nature of the results we describe and the form they might take as we move ``beyond freeness''.''NEWLINENEWLINEFor the entire collection see [Zbl 0968.00034].