Suspensions of Fat Points and Their Intersection Forms

Abstract

In this paper a fat point in ℂ² is a 0-dimensional isolated complete intersection singularity (icis) in (ℂ², 0). Such a fat point X is defined by a system of equations f = g = 0 (f and g are function germs (ℂ²,0) → (ℂ, 0)). We assume this system of equations to be generic. In particular it means that the function germs f and g define isolated curve singularities in (ℂ², 0) with possibly minimal Milnor numbers. The p-fold suspension of X is the 1-dimensional icis X ^(p) in (ℂ³, 0) defined by the equations f(x,y) + z ^p = g(x,y) = 0. The analytic type of the icis X ^(p) depends on the choice of the equations f and g. However (for a generic choice of the equations) X ^(p) is well-defined up to deformations with constant μ and μ ₁ (for an n-dimensional icis (F ⁻¹(0),0) ⊂ (ℂ^n+k,0) defined by a map-germ F = (f ₁..., f _k ) : (ℂ^n+k, 0) → (ℂ^k, 0), μ ₁ is the Milnor number of the (n + 1)-dimensional icis ((F′)⁻¹(0), 0) ⊂ (ℂ^n+k,0) defined by the map-germ F′ = (f ₁,..., f _k ) :(ℂ^n+k, 0) → (ℂ^k−1,0) for a generic choice of the equations f ₁,..., f _k−1, f _k of the icis (F ⁻¹(0),0)). Thus the Milnor fibre, the vanishing lattice, the monodromy group, (the set of) Coxeter-Dynkin diagrams,... of the suspension X ^(p) are well-defined. The relation between the monodromy operator of an icis and the monodromy operator of its p-fold suspension has been described in [ES]. The 2-fold suspension of the icis X is called its stabilization. Coxeter-Dynkin diagrams of stabilizations of fat points in ℂ², obtained by a version of the method of real morsifications, have been described in [EG2]. Here we give a similar description for p-fold suspensions.

Supported by INTAS-94-4373 and Deutsche Forschungsgemeinschaft (436 RUS 17/171/95).