Seminar in Real and Complex Geometry

Thursday, March 16, 2023, 17:15-18:45, online via zoom

Victor Vassiliev
(Weizmann Institute)

Complements of discriminants of real function singularities


Let $f: ( {\mathbb R}^n,0) \to ( {\mathbb R}, 0)$ be a smooth function with a singularity at the origin (i.e. $df(0)=0$), and $F: {\mathbb R}^n \times {\mathbb R}^l \to {\mathbb R}$ be its deformation (which can be considered as a family of functions $f_\lambda$, where $\lambda \in {\mathbb R}^l$ is parameter, $f_0 \equiv f$). The {\em discriminant variety} of such a deformation is the set of parameters $\lambda$ such that $f_\lambda$ has a critical point with zero critical value. For a generic deformation, this set is a hypersurface in the parameter space, dividing it into several local connected components. The enumeration of these components is a variation of the problem of real algebraic geometry on rigid isotopy classification of non-singular algebraic hypersurfaces: it differs from the classical problem by the function space, equivalence relation, and "boundary conditions" imposed by the original singular function.
In the case of simple singularities $A_k$, $D_k$, $E_6$, $E_7$, $E_8$, E.Looijenga has proved in 1978 a one-to-one correspondence between these components and conjugacy classes of involutions with respect to eponimous reflection groups. I will give an explicit enumeration of these components for simple singularities (which therefore also gives an enumeration of these conjugacy classes), and for the next in difficulty class of {\em parabolic} singularities. Also I will describe a combinatorial algorithm for searching and enumerating such components for arbitrary isolated singularities.