Marked ideals and their support

Let $X$ be a smooth $K$-scheme of pure dimension $n$ (Conventions for the resolution development) and let $E$ be a finite family of reduced divisors on $X$ in simultaneous SNC position (Simple normal crossings divisors and simultaneous normal crossings position). A marked ideal on $X$ is a triple $(\mathcal{I},E,\mu )$ consisting of a coherent ideal sheaf $\mathcal{I}\subseteq {\mathcal{O}}_X$, the family $E$, and an integer $\mu \ge 0$; it is displayed as $(\mathcal{I},\mu )$ when $E$ is understood. The support of $(\mathcal{I},E,\mu )$ is $\mathrm{supp}(\mathcal{I},E,\mu ):=\{x\in X:{\mathrm{ord}}_x(\mathcal{I})\ge \mu \}$ (Order of an ideal sheaf at a point); it is in fact a closed subset of $X$, as shown below. The canonical-resolution existence theorem below requires $\mu \ge 1$ and $\mathcal{I}$ nonzero at the generic point of every irreducible component of $X$. Merely requiring $\mathcal{I}\mathrm{\ne }0$ globally does not suffice on a disconnected smooth scheme. For $\mu =0$ the support is all of $X$, so support-clearing resolution by the canonical algorithm is not asserted. Writing $(\mathcal{I},\mu )$ for $(\mathcal{I},E,\mu )$ suppresses only $E$, never the order $\mu$.