Algebraic geometry by Dieudonne J.

By Dieudonne J.

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We remark that if corank df(O) = 1 (the complete intersection is equivalent to a hypersurface), then ni C-+ Cis an isomorphism and J ~ C. 6). This fact is generalized by the following conjecture ofK. Saito. Conjecture. The germ of the complement to the discriminant in the base of a contact-versal deformation of a complete intersection of positive dimension is a K(n, 1) space. The requirement that the dimension of the complete intersection to be deformed be positive is essential: Knörrer (1982) proved that the complement to the corresponding discriminant of the multiple point xi = x~ = 0 in

Semicycles vanish, and hence, J1. homology classes of WfW 0 . The corresponding anti-invariant cycles in the homology of the two-sheeted cover of W which is branched along W 0 also forms a distinguished (in the new sense) basis of the J1. short cycles in H-. For the singularities A", D", E" this construction Ieads to the usual Dynkin diagrams. But already for B" we obtain unusual ones: in the symmetric case the diagram splits into points, andin the skew-symmetric case it is a complete graph with double edges.

7, 8, 9. 8. Parabolic and Hyperbolic Singularities (Ebeling (1987a, 1987b)). Here we only consider singularities of even dimension m > 0. Their intersection index is a quadratic form. o the dimension of its kernel. Set e = (-l)m/2. Definition. -. = 1. 32 Chapter 1. Classification of Functions and Mappings m=1: • ~+q+r -2 P+Cl-1~ p+q +r+s -4k-1 /~ ~q -2 q ·.. :: p+Q. - 1 .. <. _-"'' ~ k-3 Fig. 8. • (k = 2m + p + q + r + s - 4) Examples. a) The singularity jjm+ 3 is parabolic since its lattice has decompositions into direct ( , )-orthogonal sums: fi = H l.

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