Tangent cones to discriminant loci for families of hypersurfaces

摘要: A deformation of a variety with (nonisolated) hypersurface singularities, such as projective or theta divisor an abelian variety, determines rational map the singular locus to space and resulting geometry describes how singularities propagate in deformation. The basic principle is that model dual tangent cone discriminant detailed study method, which emerged from interpreting Andreotti-Mayer's work on divisors terms Schlessinger's theory given along examples, applications, multiplicity formulas. Introduction. This paper about associated deformations nonisolated singularities. "hypersurface" whose projectivized discriminant. includes infinitesimal form Bertini's theorem for linear system converse condition double points divisors. Now we describe main results more precisely, working category complex analytic spaces. Let X be can locally written nonsingular let -'* S = ir-1(0). Then T1, sheaf first order X, line bundle sg.X, ir Kodaira-Spencer To(S) -k H0(T1), i.e. locus. If are this has no base corresponding morphism sg.X -* PTO (S) satisfies 0(p) [ir*,p(TpX)] p E hence called Gauss ir. For degree d, d-fold Veronese restricted locus, 0 assigning point its quadric cone. Assuming restriction critical C proper, consider D ir(C) reduced structure. In addition above, suppose finally expected codimension X. DUALITY THEOREM. There finite collection 8ubvarietie8 {ZZ,} intrinsic irreducible components, Received by editors March 31, 1987. contents were presented October 18, 1986 author special session algebraic at Charlotte, North Carolina, meeting AMS. 1980 Mathematics Subject Classification (1985 Revision). Primary 14D15, 14N05; Secondary 32G11, 53C57. (D1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page