Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors

Let $(\mathcal{I},E,\mu )$ be a marked ideal with $\mathcal{I}\mathrm{\ne }0$ on the smooth $K$-scheme $X$ (Marked ideals and their support). It is of maximal order if ${\mathrm{ord}}_x(\mathcal{I})\le \mu$ for every $x\in X$; when $\mu$ is attained this is equivalent to $\mathrm{supp}(\mathcal{I},E,\mu )=\{x:{\mathrm{ord}}_x\mathcal{I}=\mu \}$. For $\mu \ge 1$, in characteristic zero or in perfect characteristic $p>0$ with $\mu <p$, maximal order is also equivalent to ${\mathcal{D}}^{\mu }(\mathcal{I})={\mathcal{O}}_X$ (Field, Iterated derivative ideals preserve support in the safe characteristic range). For the tangent-direction construction require $\mu \ge 1$. Put $T(\mathcal{I}):={\mathcal{D}}^{\mu -1}(\mathcal{I})$ (Derivative ideals of an ideal sheaf and of a marked ideal). A tangent direction of $(\mathcal{I},E,\mu )$ on an open $U\subseteq X$ is a section $u\in T(\mathcal{I})(U)$ of multiplicity one (Order of an ideal sheaf at a point): ${\mathrm{ord}}_x(u)=1$ for every $x\in V(u)$, so $V(u)$ is a regular hypersurface in $U$ containing $\mathrm{supp}(\mathcal{I},E,\mu )\cap U$. Such a $u$ is transversal to $E$ at $x$ if $x\in V(u)$ and the class of $u$ together with the classes of the distinct local boundary equations through $x$ is linearly independent in ${\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2$, equivalently these equations and $u$ extend together to a regular system of parameters of ${\mathcal{O}}_{X,x}$ (Simple normal crossings divisors and simultaneous normal crossings position); in particular $u$ cannot be a boundary parameter or lie in the span of the boundary classes. This hypothesis ensures that $V(u)$ is transversal to the boundary and is used in restriction, completion-automorphism and glueing arguments.