Image Milnor number and ${\mathcal{A}}_e$-codimension for maps between weighted homogeneous irreducible curves

Abstract

Let $(X,0)\subset ({\mathbb{C}}^n,0)$ be an irreducible weighted homogeneous singularity curve and let $f:(X,0)\to ({\mathbb{C}}^2,0)$ be a map germ finite, one-to-one and weighted homogeneous with the same weights of $(X,0)$. We show that ${\mathcal{A}}_e$-$codim(X,f)={\mu }_I(f)$, where ${\mathcal{A}}_e$-$codim(X,f)$ is the ${\mathcal{A}}_e$-codimension, i.e., the minimum number of parameters in a versal deformation and ${\mu }_I(f)$ is the image Milnor number, i.e., the number of vanishing cycles in the image of a stabilisation of $f$.

17 pages