Tangent Cones to Discriminant Loci for Families of Hypersurfaces

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A deformation of a variety with (nonisolated) hypersurface singularities, such as a projective hypersurface or a theta divisor of an abelian variety, determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation. The basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation. A detailed study of the method, which emerged from interpreting Andreotti-Mayer’s work on theta divisors in terms of Schlessinger’s deformation theory of singularities, is given along with examples, applications, and multiplicity formulas.