Finite local projection of a reduced hypersurface germ

Center at $p$ by writing $\tilde{f}(z)=f(p+z)$. By [F2] choose an invertible complex-linear map $T$ with $\tilde{f}\circ T$ regular in the last variable of order $d$, and by [F3] prepare $\tilde{f}\circ T=uW$ with $W$ a Weierstrass polynomial of degree $d$. Since $f$ is a nonunit, $d>0$. Shrinking to a neighbourhood on which $u$ has no zeros, which [F4] permits, gives $Z(\tilde{f}\circ T)=Z(W)$ there; the map $z\mapsto p+Tz$ transports this local model to the original germ.

Apply [F5] to $W$ and shrink further: there are a base polydisc $V$ about $0\in {\mathbb{C}}^{n-1}$ and a radius $r>0$ such that for every $z^{\mathrm{\prime }}\in V$ the slice $T\mapsto W(z^{\mathrm{\prime }},T)$ has no zero on $\mathrm{\mid }T\mathrm{\mid }=r$ and exactly $d$ zeros in $D=\{T:\mathrm{\mid }T\mathrm{\mid }<r\}$, counted with multiplicity.

By [F6], the classes of $1,T,\dots ,T^{d-1}$ generate ${\mathcal{O}}_{n,0}\mathrm{/}(W)$ as an ${\mathcal{O}}_{n-1,0}$-module, so this quotient is finite.

The projection $\pi :Z(W)\cap (V\times D)\to V$ is surjective: for each $z^{\mathrm{\prime }}\in V$, the slice $W(z^{\mathrm{\prime }},\cdot )$ is monic of degree $d\ge 1$, hence has a root by [F10], and all its roots lie in $D$ by step 2.1. Each fibre is finite, with at most $d$ points by [F10].

Let $z_0^{\mathrm{\prime }}\in V$ with $D_W(z_0^{\mathrm{\prime }})\mathrm{\ne }0$. By [F9], $W(z_0^{\mathrm{\prime }},\cdot )$ has $d$ distinct roots ${\tau }_1,\dots ,{\tau }_d$, each simple. For each root [F7] gives a local holomorphic graph $T={\phi }_k(z^{\mathrm{\prime }})$ with ${\phi }_k(z_0^{\mathrm{\prime }})={\tau }_k$. Intersecting the finitely many base neighbourhoods and shrinking so the differences ${\phi }_k-{\phi }_l$ remain nonzero gives a common neighbourhood $U$ on which the graphs are defined and pairwise disjoint.

For compact $K\subseteq V$, let $E_K:=\{(z^{\mathrm{\prime }},T)\in K\times \bar{D}:W(z^{\mathrm{\prime }},T)=0\}$. Continuity of $W$ makes $E_K$ closed in the compact set $K\times \bar{D}$. By step 2.1 no slice has a zero on $\mathrm{\partial }D$, so $E_K={\pi }^{-1}(K)$ for $\pi :Z(W)\cap (V\times D)\to V$. Therefore ${\pi }^{-1}(K)$ is compact and $\pi$ is proper.

For $z^{\mathrm{\prime }}\in U$ the degree-$d$ polynomial $W(z^{\mathrm{\prime }},\cdot )$ vanishes at the $d$ distinct points ${\phi }_1(z^{\mathrm{\prime }}),\dots ,{\phi }_d(z^{\mathrm{\prime }})$ from step 3.2, so these are all its roots by [F10]. Thus ${\pi }^{-1}(U)={\bigcup }_{k=1}^d\{(z^{\mathrm{\prime }},{\phi }_k(z^{\mathrm{\prime }})):z^{\mathrm{\prime }}\in U\}$ is a disjoint union of graphs, each mapped biholomorphically onto $U$.

By [F8] the discriminant is not the zero germ, and by [F9] its complement is exactly the set of base points with distinct roots. For each such point step 4.1 gives a neighbourhood with $d$ disjoint sheets, so the restriction of $\pi$ over $V\setminus \{D_W=0\}$ is a $d$-sheeted holomorphic covering as defined in [F11].

If $n=1$, the base $V$ is a point. Steps 3.1 and 3.3 give surjectivity, finite fibres and properness; step 2.2 gives the finite quotient module. By [F8] and [F9] the reduced one-variable polynomial has nonzero discriminant and therefore $d$ distinct roots, so its finite zero set is a $d$-sheeted covering of the point.