A BlowUpSequence X is a finite succession of monoidal transformations starting at X, defined inductively: nil X is the empty succession, and cons X D S' is the blow-up of X with centre D followed by a succession S' starting at D.blowUp. From it one reads off

• the length r (length),

• the stages X_0 = X, X_1, \dots, X_r (stage i, indexed by Fin (r + 1)),

• the centres D_i \subseteq X_i for i < r (center i, an ideal sheaf on stage i.castSucc),

• the partial composites \Pi_i \colon X_i \to X (stageMap i),

• the last stage X_r (last) and the composite \Pi \colon X_r \to X (composite).

The centres are arbitrary closed subschemes: empty and trivial centres are allowed [Kol07, Warning 20], and the main theorems state the conditions they ask of the centres. Because the stages are defined by recursion, a centre lives on a stage whose type depends on the succession; the statements index the stages by Fin (S.length + 1) and the centres by Fin S.length, with i.castSucc the stage on which the i-th centre lies and Fin.last S.length the last stage.

Along a succession S, by recursion on the stages:

• S.weakTransformSeq J i is Hironaka's sequence of weak transforms: J_0 = J and J_{i+1} the weak transform of J_i along the i-th blow-up (Definition 2.2.5.2);

• S.boundarySeq E i is Hironaka's sequence of boundaries: E_0 = E and E_{i+1} = \mathrm{red}(f_i^{-1}(E_i) \cup f_i^{-1}(D_i));

• S.strictTransformSeq Y i is the strict transform Y_i of a closed subscheme Y [Kol07, Definition 30.2]; Włodarczyk's Y_i [Wlo05, Theorem 1.0.2 (b)].

Kollár's divisor E = \sum E^i with an ordered index set: a DivisorFamily X is a finite linearly ordered index type ι (with its fintype and linearOrder) and a closed subscheme component i for each index, given by its ideal sheaf. The order matters to Kollár's algorithm, in which new exceptional divisors come after old ones. DivisorFamily.empty X is the empty divisor; support is the set of points on at least one component and singularLocus (\operatorname{Sing} E) the set of points on at least two; E.comap h is the inverse image family h^{-1}(E) [Kol07, 34.1].

Kollár's simple normal crossings, illustrated with Hironaka's notion in Normal crossings. At a point x, E.IsSncAt x z says that z_1, \dots, z_n is a regular system of parameters of \mathcal{O}_{X,x} and each component of E through x is cut out near x by its own coordinate z_{c(i)}, distinct components by distinct coordinates. E.IsSnc asks every component to be smooth and such coordinates to exist at every point [Kol07, Definition 24 (1)–(3)]; since the coordinates are asked for also at the points on no component, E.IsSnc makes X itself regular. E.HasSncWith Z says that at every point of Z the coordinates can be chosen so that in addition Z is cut out by some of them [Kol07, Definition 24 (4)]; some components may contain Z.

Divisors with simple normal crossings, as ideal sheaves and as sets. IsIdealOfSncDivisor I says that I is the ideal sheaf of an effective divisor \sum a_j F_j supported on a simple normal crossings family F (Definition 2.4.2.2); Kollár's principalization makes the pull-back of an ideal sheaf of this form [Kol07, Theorem 35 (2)]. IsSncDivisor Z says that the set Z is the support of a simple normal crossings divisor family [Kol07, Definition 24], with its reduced structure; a strong resolution asks this of the preimage of the singular locus [Kol07, (3)].

Kollár's total transform of a divisor family under one blow-up with centre D is the family of the birational transforms of the components of E, followed by the exceptional divisor f^{-1}(D) with the largest index (DivisorFamily.totalTransform, [Kol07, Definitions 25 and 65]). Along a succession it gives E_i = (\pi_0 \cdots \pi_{i-1})^{-1}_*(E) + \mathrm{Ex}_{tot}(\pi_0 \cdots \pi_{i-1}) (BlowUpSequence.totalTransformSeq); starting from the empty family it is the family of exceptional divisors, as in the embedded desingularization theorem (Theorem 4.2.10.4).

These operations compare successions on different schemes:

• S.pullback h is the pull-back h^* S along h \colon Y \to X [Kol07, Definition 30.1]: blow up the inverse images of the centres, using the induced morphisms of blow-ups (Definition 2.2.3.1); for smooth (more generally flat) h the stages are X_i \times_X Y;

• S.eraseEmpty deletes every blow-up whose centre is empty and reindexes the rest [Kol07, 32 and 34.1], carrying the tail back along the isomorphism of the empty blow-up;

• S.pushforward j is the push-forward j_* S of a succession on a closed subscheme along the closed immersion j [Kol07, Definition 30.3]: the centres viewed in the ambient stages (Scheme.Hom.pushforwardBlowUp, Definition 2.2.5.1).

A pulled-back succession may have empty centres where the image of h misses a centre, which is why, for a non-surjective h, the functorial theorems compare with (S.pullback h).eraseEmpty.

HasUnderlyingScheme C k equips a category of inputs C with a functor forget to AlgScheme k; scheme X is the underlying scheme of an input and hom h the underlying morphism of schemes of a morphism of inputs. A blow-up sequence functor BlowUpSequenceFunctor C assigns to every input X a succession of blow-ups B X : BlowUpSequence (scheme X) of its underlying scheme [Kol07, Definition 31]. It carries no conditions: the conditions on the values, and the compatibility with smooth morphisms (Definition 4.2.4.3), are asserted by the main theorems, which quantify over the input so that a single functor works for all inputs, and over the field so that a single family of functors works for all fields.