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

Abstract

<jats:title>Abstract</jats:title> <jats:p>Let (<jats:italic>X</jats:italic>, 0) ⊂ (ℂ<jats:sup><jats:italic>n</jats:italic></jats:sup>, 0) be an irreducible weighted homogeneous singularity curve and let <jats:italic>f</jats:italic> : (<jats:italic>X</jats:italic>, 0) → (ℂ<jats:sup>2</jats:sup>, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (<jats:italic>X</jats:italic>, 0). We show that 𝒜<jats:sub><jats:italic>e</jats:italic></jats:sub>-codim(<jats:italic>X</jats:italic>, <jats:italic>f</jats:italic>) = <jats:italic>μ<jats:sub>I</jats:sub></jats:italic>(<jats:italic>f</jats:italic>), where the 𝒜<jats:sub><jats:italic>e</jats:italic></jats:sub>-codimension 𝒜<jats:sub><jats:italic>e</jats:italic></jats:sub>-codim(<jats:italic>X</jats:italic>, <jats:italic>f</jats:italic>) is the minimum number of parameters in a versal deformation and <jats:italic>μ<jats:sub>I</jats:sub></jats:italic>(<jats:italic>f</jats:italic>) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of <jats:italic>f</jats:italic>.</jats:p>