Higher discriminants and the topology of algebraic maps

Given a map between algebraic varieties, we define loci in the target of the map which we name the “higher discriminants.” They are defined in terms of transversality conditions, and, in case of a map between smooth varieties, they can be determined by a tangent space calculation. We prove that the higher discriminants control the variation of cohomology of the fibers in the following two senses: (1) the support of any summand of the pushforward of the intersection cohomology sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the proper pushforward of the constant function is a conormal variety to a component of a higher discriminant. As an example, in the last section of the paper, we show that in the case of an algebraic completely integrable system, the stratification by higher discriminants gives exactly the δ-stratification introduced by Ngˆo