Order functions and normal-crossings strata are upper semicontinuous

The order function is upper semicontinuous and the derivative formula holds in the stated range. The superlevel set for $k$ is closed by [F1] in every characteristic. If $\mathrm{char}K=0$ or $p>k$, the formula with $V({\mathcal{D}}^{k-1}(\mathcal{I}))$ is [F1]. For the remaining safe endpoint $p=k$, first suppose ${\mathrm{ord}}_x(\mathcal{I})\ge k$. Every coordinate derivative of order $r\le k-1$ lowers order by at most $r$, so ${\mathcal{D}}^{k-1}(\mathcal{I}{)}_x\subseteq {\mathfrak{m}}_x$ and $x\in V({\mathcal{D}}^{k-1}(\mathcal{I}))$. If instead $j:={\mathrm{ord}}_x(\mathcal{I})<k$, choose $f\in {\mathcal{I}}_x$ with order $j$ and a nonzero monomial $c{U}^{\alpha }$ in its degree-$j$ initial form. Then $\mathrm{\mid }\alpha \mathrm{\mid }=j\le k-1$ and ${\mathrm{\partial }}^{\alpha }f$ has nonzero residue $c\alpha !$ in $\kappa (x)$, since every factor in $\alpha !$ is less than $p=k$. This derivative lies in ${\mathcal{D}}^{k-1}(\mathcal{I}{)}_x$, so that ideal is a unit at $x$ and $x\mathrm{\notin }V({\mathcal{D}}^{k-1}(\mathcal{I}))$. Thus the formula holds also for $p=k$, proving assertion (1).

The normal-crossings count is upper semicontinuous. For a finite set $J$ of members of $E$ the set $\{x:x\in D\text{ for all }D\in J\}={\bigcap }_{D\in J}D$ is closed by [F2], and $\{x:s_E(x)\ge k\}$ is the finite union of these intersections over the $k$-element subsets $J$; hence it is closed, and the function is upper semicontinuous. On the locally closed stratum where exactly the members of $J$ pass through the point, $s_E$ is constant equal to $\mathrm{\mid }J\mathrm{\mid }$; these finitely many strata cover $X$ locally because $E$ is finite and its members have SNC, so only finitely many subsets occur locally at a point [F2]. This is assertion (2).

Work on an open neighbourhood where $f,g$ have finite ranges; the order functions satisfy this bound by [F4]. For a lexicographic threshold $(a,b)$, the superlevel set is $\{f>a\}\cup (\{f\ge a\}\cap \{g\ge b\})$. Since $f$ has finite range locally, $\{f>a\}$ is a finite union of closed superlevel sets, and the second set is closed by upper semicontinuity. Thus this union is closed. Induction gives the same result for a finite tuple; the superlevel set of a finite maximum is the finite union of the corresponding closed sets. This proves (3). It makes no claim that an unspecified piecewise assembly has satisfied these hypotheses.