  We show that the way in which Betti cohomology varies in a proper family of
complex algebraic varieties is controlled by certain "higher discriminants" in
the base. These discriminants are defined in terms of transversality
conditions, which in the case of a morphism between smooth varieties can be
checked by a tangent space calculation. They control the variation of
cohomology in the following two senses: (1) the support of any summand of the
pushforward of the IC 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.
  The same would hold for the Whitney stratification of the family, but there
are vastly fewer higher discriminants than Whitney strata. For example, in the
case of the Hitchin fibration, the stratification by higher discriminants gives
exactly the {\delta} stratification introduced by Ngo.
