A remark on isolated complex hypersurface singularities

Abstract

This is now an expository note about the following classical problem. Let $(X,0)$ be the germ of a hypersurface in $({\mathbb{C}}^n,0)$ with an ordinary singularity of multiplicity $m$ at the origin $0$. A natural question to ask is whether $X$ and its tangent cone at the origin are analytically isomorphic. The answer is negative in general, in view of a theorem of Kioji Saito. However there is an integer $D(n,m)>m$ such that, given a \emph{regular} homogeneous polynomial $f(x_1,\dots ,x_n)$ of degree $m$ (this means that $\{f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\ge D(n,m)$, any convergent power series of the form $g=f+o(d)$ (here, as usual, $o(d)$ stays for a power series of order at least $d$), defines a germ $\{g=0\}$ which is analytically equivalent to the germ $\{f=0\}$. In this note we compute $D(n,m)$ explicitly as $n(m-2)+1$. We also give an extension to the case in which $f$ is a quasihomogeneous polynomial. It was pointed out that the value of $D(n,m)$ was already known by \cite[Exercise 7.31]{D}.

7 pages