A reduced prepared hypersurface stays reduced nearby

By [F1] the germ $W$ is regular in the last variable of order $d$ and is its own preparation, so [F2] applies to it: $D_W$ is a nonzero base germ and [F3] identifies its vanishing with the existence of a repeated root in the slice $W(z^{\mathrm{\prime }},\cdot )$. Shrink the product representative so that $W$ and $D_W$ are defined on the connected base $V$ and the finite-projection conclusions hold.

Suppose for contradiction that some $q=(q^{\mathrm{\prime }},\tau )\in Z(W)\cap (V\times D)$ has a nonreduced germ: $W_q=g^{2}h$ for an irreducible germ $g$ at $q$. By [F5] the germ $g$ is a nonzero nonunit, so $g(q^{\mathrm{\prime }},\tau )=0$.

The vertical slice $\zeta \mapsto g(q^{\mathrm{\prime }},\zeta )$ is not identically zero near $\tau$: the identity $W=g^{2}h$ holds on a neighbourhood of $q$, so if that slice vanished identically then the slice $\zeta \mapsto W(q^{\mathrm{\prime }},\zeta )$ would vanish identically near $\tau$, contradicting that this slice is the monic polynomial of degree $d$ from step 1.1, which has only finitely many zeros. Hence $g$ is regular in the last variable of some order $m\ge 1$ at $q$, by [F1] and the vanishing of $g$ at $q$.

Choose a product neighbourhood $U_0\times \{\mathrm{\mid }\zeta -\tau \mathrm{\mid }<{\rho }_0\}$ of $q$ contained in the neighbourhood where $W=g^{2}h$ holds, and shrink ${\rho }_0$ so the slice $g(q^{\mathrm{\prime }},\cdot )$ has no zero on $\mathrm{\mid }\zeta -\tau \mathrm{\mid }={\rho }_0$. Prepare the regular germ $g$ at $q$: $g=u_{g}G$ with $G$ a Weierstrass polynomial of degree $m\ge 1$ in the translated coordinates. Apply the stability of the slice zero count [F6] on this chosen disc and shrink the base to a neighbourhood $U\subseteq U_0$ of $q^{\mathrm{\prime }}$; every slice $g(z^{\mathrm{\prime }},\cdot )$, $z^{\mathrm{\prime }}\in U$, then has exactly $m\ge 1$ zeros counted with multiplicity in $\mathrm{\mid }\zeta -\tau \mathrm{\mid }<{\rho }_0$. In particular the product used below remains inside the factorization neighbourhood.

Fix $z^{\mathrm{\prime }}\in U$ and let $\zeta$ be one of the $m\ge 1$ zeros of $g(z^{\mathrm{\prime }},\cdot )$ supplied by step 3.1. Because the identity $W=g^{2}h$ holds on a neighbourhood of $q$, the slices satisfy $W(z^{\mathrm{\prime }},\cdot )=g(z^{\mathrm{\prime }},\cdot {)}^{2}h(z^{\mathrm{\prime }},\cdot )$ near $\zeta$; by [F7] the slice of $g$ has a zero of some order $m_y\ge 1$ at $\zeta$, so the slice of $W$ vanishes there to order at least $2m_y\ge 2$. Thus $\zeta$ is a repeated root of $W(z^{\mathrm{\prime }},\cdot )$, and [F7] gives ${\mathrm{\partial }}_{T}W(z^{\mathrm{\prime }},\zeta )=0$ while [F3] gives $D_W(z^{\mathrm{\prime }})=0$.

Every $z^{\mathrm{\prime }}\in U$ therefore lies in the zero set of $D_W$. If $n=1$, the base $V$ is the single point of ${\mathbb{C}}^0$; step 4.1 gives $D_W=0$ there, contradicting the nonzero constant $D_W$ from step 1.1. If $n\ge 2$, $D_W$ vanishes on the nonempty open set $U$, contradicting [F4] on the connected base $V$ because $D_W$ is the nonzero base germ from step 1.1. Hence no point $q\in Z(W)\cap (V\times D)$ has a nonreduced local germ.

Shrinking to the product representative $V\times D$ fixed in step 1.1, every local equation germ of $W$ at a point of its zero set is reduced, which is the assertion.