A'Campo's theorem on the discriminant (Q1896952)

The Dynkin diagrams of singularities can be trees only for simple critical points. This result by N. A'Campo, has not been advanced whatsoever in the direction of increase of modality, sharing the fate of many beautiful theorems on simple \(A - D - E\) singularities. Counterexamples to naive conjectures were constructed by W. Ebeling. In the present paper we also remain in the modality 0. A'Campo's theorem and a theorem close to it stating that only the standard Dynkin diagrams of simple singularities contain no edges of negative weight are generalized to degenerate decompositions of a singularity that appear in the base of its bounded miniversal deformation. We also prove A'Campo's theorem for boundary singularities and a block version of the formula representing the Coxeter operator as a product of upper and lower triangular matrices (in the terminology of the theory of singularities, this is a formula relating the variation and the classical monodromy operators). Moreover, for real deformations of singularities, we prove a formula relating the classical monodromy operator to the intersection form; some conjectures on monodromy of singularities are also stated.