Image Milnor number and 𝒜 e -codimension for maps between weighted homogeneous irreducible curves

Abstract Let (X, 0) ⊂ (ℂ n , 0) be an irreducible weighted homogeneous singularity curve and let f : (X, 0) → (ℂ2, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (X, 0). We show that 𝒜 e -codim(X, f) = μI (f), where the 𝒜 e -codimension 𝒜 e -codim(X, f) is the minimum number of parameters in a versal deformation and μI (f) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of f.