ON THE LEGACY OF FREE DIVISORS - DISCRIMINANTS AND MORSE-TYPE SINGULARITIES

We investigate when the freeness of a divisor (V, 0) is inherited by t he discriminants for the versal deformations of nonlinear sections of V. We introduce Morse-type singularities for sections and give a crite rion for freeness of the discriminant in terms of (V, 0) generically h aving Morse-type singularities. This criterion is applied to determine when the bifurcation sets of mappings and smoothings of space curves and complete intersections are free. It also explains the failure of f reeness for discriminantal arrangements of hyperplanes.