Complex-analytic hypersurface germ and its reduced equation

Fix $n\ge 1$ and $p\in {\mathbb{C}}^n$.

Set germs. Two subsets $S,S^{\mathrm{\prime }}$ of neighbourhoods of $p$ define the same set germ at $p$ when $S\cap W=S^{\mathrm{\prime }}\cap W$ for some neighbourhood $W$ of $p$ contained in both domains. A set germ is written $(S,p)$, and containment of set germs is defined by containment of suitable representatives. This is the standard equivalence relation of germs of sets; it is the analogue for subsets of the equivalence of holomorphic functions used for the germ ring ${\mathcal{O}}_{{\mathbb{C}}^n,p}$ (The ring of holomorphic germs at $0$ and its maximal ideal).

Hypersurface germs. A nonempty proper set germ $X$ at $p$ is a complex-analytic hypersurface germ at $p$ when there is a nonzero nonunit germ $f\in {\mathcal{O}}_{{\mathbb{C}}^n,p}$ with

$X=(Z(f),p),Z(f)=\{z:f(z)=0\},$

the zero set of a representative of $f$ near $p$. Every nonzero nonunit produces a nonempty proper zero germ: $f(p)=0$ because nonunits are exactly the germs vanishing at the base point (A germ is a unit exactly when its value at $0$ is nonzero, so ${\mathcal{O}}_{m,0}$ is local), and $Z(f)$ is not all of a neighbourhood of $p$ because a nonzero germ is not identically zero on any neighbourhood.

Reduced defining germ. Let $X=(Z(f),p)$ be a hypersurface germ and let $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ be the square-free reduction of $f$ (Square-free reduction of a holomorphic equation), so $Z(f_{\mathrm{r}\mathrm{e}\mathrm{d}})=Z(f)$ and $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is reduced (Reduced holomorphic germ for a hypersurface). Then $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is called the reduced defining germ of $X$, and $f$ is called a defining equation of $X$.

Well-definedness. If $f^{\mathrm{\prime }}$ is any other nonzero nonunit with $(Z(f^{\mathrm{\prime }}),p)=X$, then $f^{\mathrm{\prime }}$ vanishes on $Z(f_{\mathrm{r}\mathrm{e}\mathrm{d}})$ near $p$ and $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ vanishes on $Z(f^{\mathrm{\prime }})$ near $p$, so the two reduced germs lie in the same vanishing ideal:

$f_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}\in I_p(X)=(f_{\mathrm{r}\mathrm{e}\mathrm{d}})\text{and}{f}_{\mathrm{r}\mathrm{e}\mathrm{d}}\in (f_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}),$

by the principal vanishing-ideal lemma (The vanishing ideal of a reduced hypersurface germ is principal). Hence $f_{\mathrm{r}\mathrm{e}\mathrm{d}}^{\mathrm{\prime }}$ is a unit multiple of $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$: two reduced defining germs of the same hypersurface germ differ by a unit of the germ ring. The reduced defining germ is therefore determined by $X$ up to a unit, and since a set germ is independent of the chosen representative neighbourhood, the hypersurface germ and its reduced defining germ are geometric objects attached to $X$ and not to a particular equation or neighbourhood.