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In [13] the technique of semi-simplicial resolutions was applied to the study of the topology of the image of a stable perturbation / of a map-germ C" —> C with n <p. The crucial role was played there by the multiple point sets D(f). These are the closures, in (C)*, of the sets of /c-tuples of pairwise distinct points sharing the same image under / The set D is invariant under the action of the permutation group Sk, interchanging the copies of the source. According to [13], there exists a spectral sequence computing the homology of the image Y of f whose E term is formed by the homology of the sets D, anti-symmetric (alternating) with respect to these actions. We get a good situation starting with a corank-1 map-germ of finite s&codimension. Then each D(f) is a Milnor fibre of an isolated complete intersection singularity (icis) and, thus, has non-trivial alternating homology only in one dimension, at least over <Q. As a consequence, the spectral sequence mentioned degenerates at E and the rational homology of Y is the sum of the alternating homology of the D [13]. This decomposition is very useful in the study of the mixed Hodge structure [13], of the monodromy of the image [12], etc. For maps C » C P with n^p, the image should be replaced by the discriminant, that is, by the set of the critical values of the mapping. Indeed, the discriminant hypersurface is very much like the image of a map C~-+C. For example, if we are in the nice dimensions [4], the discriminant of a perturbation of a map-germ of a finite j^-codimension has the homotopy type of a wedge of a finite number of middle-dimensional spheres [8], that is, we get the same as for the image mentioned [18]. In the present paper we introduce the geometric approach to the algebraic construction of the semi-simplicial resolution used in [13]. We extend the results, obtained in [13] for rational homology, to the case of integer coefficients. And we apply these results to the investigation of the topology of discriminants of corank-1 map-germs. In § 1 we give a description of a geometric realization of the semi-simplicial resolution of the image of a finite map / . We show that the homological spectral sequence, associated to the natural filtration on this resolution, has E term consisting of the homology Alt H+ of the alternating chain subcomplexes of the multiple point sets D(f) of/.

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