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

TL;DR

This paper establishes a precise relationship between the $ ext{A}_e$-codimension and the image Milnor number for weighted homogeneous maps from irreducible singularity curves, providing a new invariant connection in singularity theory.

Contribution

It proves that for weighted homogeneous irreducible curve singularities, the $ ext{A}_e$-codimension equals the image Milnor number, linking deformation parameters to topological invariants.

Findings

$ ext{A}_e$-codimension equals the image Milnor number for the class studied.

Provides a formula connecting deformation theory and topological invariants.

Enhances understanding of singularity invariants in weighted homogeneous settings.

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 $\mathscr{A}_e$-$codim(X,f)=\mu_I(f)$, where $\mathscr{A}_e$-$codim(X,f)$ is the $\mathscr{A}_e$-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 $f$.