On the legacy of free divisors. II. Free* divisors and complete intersections

We provide a criterion that for an equivalence group $\mathcal G$ on holomorphic germs, the discriminant of a $\mathcal G$-versal unfolding is a free divisor. The criterion is in terms of the discriminant being Cohen–Macaulay and generically having Morse-type singularities. When either of these conditions fails, we provide a criterion that the discriminant have a weaker free* divisor structure. For nonlinear sections of a free* divisor $V$, we obtain a formula for the number of singular vanishing cycles by modifying an earlier formula obtained with David Mond and taking into account virtual singularities.