A regular hyperplane has a one-sheeted projection

Given: An integer $n\ge 1$, the coordinate hyperplane $X=\{z_n=0\}\subseteq {\mathbb{C}}^n$, its equation germ $z_n\in {\mathcal{O}}_{{\mathbb{C}}^n,0}$, and the projection $\pi (z)=z^{\mathrm{\prime }}$ forgetting the last coordinate.

A hypersurface germ at $0$ is the zero germ of a nonzero nonunit; its reduced defining germ is unique up to a unit (Complex-analytic hypersurface germ and its reduced equation).

A reduced germ is a nonzero nonunit that is not divisible by the square of an irreducible germ; irreducible means not a product of two nonunits, and an irreducible germ is reduced (Reduced holomorphic germ for a hypersurface, Irreducible and prime elements of an integral domain).

A Weierstrass polynomial of degree $1$ in the last variable has the form $z_n+a_0(z^{\mathrm{\prime }})$ with $a_0\in {\mathcal{O}}_{{\mathbb{C}}^{n-1},0}$, $a_0(0)=0$; in particular $z_n$ itself is a degree-one Weierstrass polynomial and $W(0,z_n)=z_n$ (Weierstrass polynomials in the last variable).

A point $q$ of a reduced hypersurface germ is regular exactly when the differential of a local reduced equation at $q$ is nonzero, equivalently exactly when the germ is a holomorphic hypersurface graph near $q$ (Regular and singular points of an analytic hypersurface).

The discriminant of a monic degree-one polynomial $t+a_1$ is $1$; in particular ${\mathrm{Disc}}_{z_n}(z_n)=1\mathrm{\ne }0$ (The discriminant of a monic polynomial as the coefficient expression of ${\mathrm{\Delta }}_n^2$).

For a prepared equation $W$ on the chosen product neighbourhood $V\times D$ of the finite projection theorem, containing all slice roots in $D$ and none on $\mathrm{\partial }D$, put $X_W:=Z(W)\cap (V\times D)$ and let ${\pi }_W:X_W\to V$ be the restricted coordinate projection. The discriminant definition and finite projection theorem give $D_W=\mathrm{Disc}(W)$, branch set $B_{{\pi }_W}=\{D_W=0\}\subseteq V$, and a proper surjection with finite fibres that is a covering with as many sheets as the degree of $W$ over $V\setminus B_{{\pi }_W}$; when $n=1$ the base is a single point (Discriminant and branch set of a fixed Weierstrass projection, Finite local projection of a reduced hypersurface germ).

Assume the Axiom of Choice. For $m\ge 0$ the Krull dimension of the holomorphic germ ring is $\dim {\mathcal{O}}_{{\mathbb{C}}^m,0}=m$, with ${\mathcal{O}}_{{\mathbb{C}}^0,0}=\mathbb{C}$; this is the only place where the Axiom of Choice is used in the present example (Krull dimension of the holomorphic germ ring, The Axiom of Choice).

The local dimension of a hypersurface germ is ${\dim }_{0}X=\dim {\mathcal{O}}_{{\mathbb{C}}^n,0}\mathrm{/}(f_{\mathrm{r}\mathrm{e}\mathrm{d}})$ for any reduced equation $f_{\mathrm{r}\mathrm{e}\mathrm{d}}$ of $X$ (Local Krull dimension of a hypersurface germ).

A germ $h\in {\mathcal{O}}_{{\mathbb{C}}^n,0}$ expands as a convergent power series in $z_n$ with coefficients $h_k\in {\mathcal{O}}_{{\mathbb{C}}^{n-1},0}$, so $h-h_0\in (z_n)$ and the substitution $z_n=0$ induces an isomorphism ${\mathcal{O}}_{{\mathbb{C}}^n,0}\mathrm{/}(z_n)\to {\mathcal{O}}_{{\mathbb{C}}^{n-1},0}$, $h+(z_n)\mapsto h(z^{\mathrm{\prime }},0)$ (A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc).

Proof technique: direct — identify the reduced equation, compute the gradient and the discriminant, and compute the local ring by expanding in the last variable.