Refined maximal-contact statement via the coefficient ideal

The support stays a proper subset of the hypersurface in every characteristic. By [F1], $\mathrm{supp}({\mathcal{I}}_i,\mu )\subseteq V(u{)}_i$ for every $i$. At $i=0$ this inclusion is proper because the support has codimension at least two throughout $U$ while $V(u)$ is a hypersurface [F4]. If it is proper at stage $i$, then $V(u{)}_i\setminus \mathrm{supp}({\mathcal{I}}_i,\mu )$ is a nonempty open subset disjoint from the next center, since every center lies in the marked support. The blow-up is an isomorphism over this open set, and the controlled transform agrees there with the unchanged ideal, so this nonempty open subset persists inside $V(u{)}_{i+1}$ and remains outside $\mathrm{supp}({\mathcal{I}}_{i+1},\mu )$. Thus the inclusion is proper at every stage.

Assume now that $K$ has characteristic zero or perfect characteristic $p>\mu$. Since $V(u)$ is a regular hypersurface containing the initial support and is not contained in that support, it satisfies the hypotheses of [F2]. Applying [F2] to $S=V(u)$ proves that $(V(u{)}_i)$ is a multiple test blow-up of $C(\mathcal{I},\mu ){\mathrm{\mid }}_{V(u)}$ and gives the support identity of clause (3) at every stage.

Under AC and the same characteristic condition, the converse clause of [F2] says that every multiple test blow-up of $C(\mathcal{I},\mu ){\mathrm{\mid }}_{V(u)}$ is induced by a multiple test blow-up of $(\mathcal{I}{\mathrm{\mid }}_U,\mu )$ with centers in the strict transforms of $V(u)$. The equality in clause (3) follows at every stage from [F2].