A remark on isolated complex hypersurface singularities
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908993008435200 |
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| author | Catanese, Fabrizio Ciliberto, Ciro Galati, Concettina |
| author_facet | Catanese, Fabrizio Ciliberto, Ciro Galati, Concettina |
| contents | This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. 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 $f(x_1,\ldots, x_n)$ of degree $m$
(this means that $\{ f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\geq D(n,m)$, 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14729 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A remark on isolated complex hypersurface singularities Catanese, Fabrizio Ciliberto, Ciro Galati, Concettina Algebraic Geometry This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. 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 $f(x_1,\ldots, x_n)$ of degree $m$ (this means that $\{ f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\geq D(n,m)$, 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}. |
| title | A remark on isolated complex hypersurface singularities |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2604.14729 |