Multiple test blow-ups, controlled transforms and resolutions of marked ideals

Let $(\mathcal{I},E,\mu )$ be a marked ideal on the smooth $K$-scheme $X$ (Marked ideals and their support). A multiple test blow-up of $(\mathcal{I},E,\mu )$ is a finite sequence $X=X_0\stackrel{{\sigma }_1}{\leftarrow }{X}_1\leftarrow \cdots \leftarrow X_r$ in which each ${\sigma }_i\text{\hspace{0.17em}⁣}:X_i\to X_{i-1}$ is specified either as an inserted isomorphism step or as the blowup (Blowup of a scheme along an ideal sheaf) of $X_{i-1}$ at a regular closed subscheme $C_{i-1}\subseteq \mathrm{supp}({\mathcal{I}}_{i-1},E_{i-1},\mu )$ (Closed immersions of schemes) that has SNC with $E_{i-1}$, together with the marked ideals defined inductively for a blowup step by $({\mathcal{I}}_i,E_i,\mu ),{\mathcal{I}}_i:=\mathcal{I}(D_i{)}^{-\mu }\cdot {\sigma }_i^{\ast }({\mathcal{I}}_{i-1}),E_i:={\sigma }_i^{\mathrm{c}}(E_{i-1})\cup \{D_i\},$ where $D_i\subseteq X_i$ is the exceptional divisor (Exceptional subscheme of a blowup), $\mathcal{I}(D_i)$ is its invertible ideal (Invertible sheaf of cartier divisor), $\mathcal{I}(D_i{)}^{-\mu }$ its $(-\mu )$-th tensor power (Invertible sheaves, Tensor product of sheaves of modules), and ${\sigma }_i^{\mathrm{c}}(E_{i-1})$ is the family of strict transforms (Strict transform of a closed subscheme) ordered so that all old members precede $D_i$. Empty members of the transformed boundary are omitted, with the order of all surviving labels retained. The transform ${\sigma }_i^{\mathrm{c}}({\mathcal{I}}_{i-1},\mu ):=({\mathcal{I}}_i,\mu )$ is the controlled transform of $({\mathcal{I}}_{i-1},\mu )$; for a local section $f\in {\mathcal{I}}_{i-1}(U)$ with local equation $y_i$ of $D_i$ the section $y_i^{-\mu }{\sigma }_i^{\ast }(f)$ is a controlled transform of $f$, well defined up to a unit. A blowup step retains the specified center and $D_i={\sigma }_i^{-1}(C_{i-1})$, even when its underlying morphism is an isomorphism, as for a Cartier center. An inserted isomorphism step has empty $D_i$, transports the ideal and ordered boundary, and appends no boundary member. A resolution of $(\mathcal{I},E,\mu )$ is a multiple test blow-up with $\mathrm{supp}({\mathcal{I}}_r,E_r,\mu )=\mathrm{\varnothing }$. An extension of a multiple test blow-up $(X_i{)}_{0\le i\le m}$ is a multiple test blow-up $(X_j^{\mathrm{\prime }}{)}_{0\le j\le m^{\mathrm{\prime }}}$ with $X_0^{\mathrm{\prime }}=X$, indices $j_0=0<j_1<\cdots <j_m$ and isomorphisms forming an identification $X_{j_i}^{\mathrm{\prime }}=X_i$; the extended sequence is obtained from the original one by inserting isomorphisms, with every original blow-up retained in its original order. No new nontrivial blow-ups are inserted. This is the extension convention of the source, Definition 2.1.5.