Finite 𝒜-determinacy of generic homogeneous map germs in ℂ³

Abstract

Denote by 𝐻(𝑑₁, 𝑑₂, 𝑑₃) the set of all homogeneous polynomial mappings 𝐹 = (𝑓₁, 𝑓₂, 𝑓₃) : ℂ³ → ℂ³, such that deg 𝑓_𝑖 = 𝑑_𝑖. We show that if gcd(𝑑_𝑖, 𝑑_𝑗) ≤ 2 for 1 ≤ 𝑖 < 𝑗 ≤ 3 and gcd(𝑑₁, 𝑑₂, 𝑑₃) = 1, then there is a non-empty Zariski open subset 𝑈 ⊂ 𝐻(𝑑₁, 𝑑₂, 𝑑₃) such that for every mapping 𝐹 ∈ 𝑈 the map germ (𝐹, 0) is 𝒜-finitely determined. Moreover, in this case we compute the number of discrete singularities (0-stable singularities) of a generic mapping (𝑓₁, 𝑓₂, 𝑓₃): ℂ³ → ℂ³, where deg 𝑓_𝑖 = 𝑑_𝑖.