A formula to calculate the invariant $J$ of a quasi-homogeneous map germ

Abstract

In this work, we consider a quasi-homogeneous, corank $1$, finitely determined map germ $f$ from $({\mathbb{C}}^2,0)$ to $({\mathbb{C}}^3,0)$. We consider the invariants $m(f(D(f))$ and $J$, where $m(f(D(f))$ denotes the multiplicity of the image of the double point curve $D(f)$ of $f$ and $J$ denotes the number of tacnodes that appears in a stabilization of the transversal slice curve of $f({\mathbb{C}}^2)$. We present formulas to calculate $m(f(D(f))$ and $J$ in terms of the weights and degrees of $f$.

12 pages, 1 figure