Topology of Discriminants and Their Complements

Abstract

1 Main definitions and examples The general notion of a discriminant is as follows. Consider any function space T, finite~dimensionai rjr not;~and som^"iuass~of~slngularitie5 S tlmt~the~functiöns" from" T can take at the points of the issue manifold. The corresponding discriminant variety E (S) C T is the space of all functions that have such singular points. For example, let T be the space of (real or complex) polynomials of the form x d + ai^" 1 + • • • + a d , (1) and S = {a multiple root}. Then (in the complex case) E (S) is the zero level set of the usual discriminant polynomial of the coefficients a$; this is a motivation for the word "discriminant" in the general situation. In the simplest nontrivial case, when d = 4, the discriminant variety in the space of real polynomials is ambient diffeomorphic to the direct product of the line R 1 and the "swallowtail", i.e. the surface shown in the lower part of Figure la. More generally, we can consider the discriminant E^ consisting of all polynomials having roots of multiplicity at least fc, or we can consider the space of polynomial systems of the form x mi +aìz mi ~ :L + ---+ aJ ni (2) :+ a k x m h -l