Let X be a smooth algebraic k-scheme, k of characteristic zero, J an ideal sheaf on X with nonzero stalks, d the maximum over the points x of the orders \nu(J_x), and E_0 a reduced closed subscheme having only normal crossings. Then there is a finite succession of monoidal transformations f_i \colon X_{i+1} \to X_i with centres D_i \subseteq X_i such that

1. every centre D_i is non-singular and irreducible;

2. the weak transform J_i of J has order at least d at every point of D_i;

3. the boundary E_i, defined by E_{i+1} = \mathrm{red}(f_i^{-1}(E_i) \cup f_i^{-1}(D_i)), has only normal crossings with D_i;

4. the last boundary E_r has only normal crossings, and the last weak transform J_r has order less than d at every point of X_r.

This is one round of order reduction: it lowers the maximal order of the ideal while keeping the accumulated exceptional divisor, together with the original boundary, a normal crossings divisor. Iterating it gives Main Theorem II(N) (Theorem 4.2.2.1). The statement includes the degenerate case d = 0, in which J = \mathcal{O}_X and condition (4) can hold only on an empty X_r; blowing up the irreducible components of X one after another achieves it, since Hironaka's monoidal transformation is empty over a centre that is a whole component.

Condition (1) is HasRegularIrreducibleCenters, and condition (3) with the first half of (4) is HasSncBoundaries (Definition 4.2.1.1). The statement also uses the order \nu(J_x) (Definition 2.1.3.2), stalks and nonzero stalks (Definition 2.1.3.1), Hironaka's normal crossings (Definition 2.1.5.1), and the weak transforms and boundaries along a succession (Definition 2.4.1.2, Definition 2.2.5.2).

Let X be a smooth algebraic k-scheme, J an ideal sheaf with nonzero stalks and E a reduced closed subscheme having only normal crossings. Then there is a finite succession of monoidal transformations with centres B_i such that

1. every centre B_i is non-singular and irreducible;

2. for each i there is a positive integer d_i which is the maximal order of the weak transform J_i and is its order at every point of B_i;

3. the boundary E_i has only normal crossings with B_i;

4. E_r has only normal crossings and J_r = \mathcal{O}_{X_r}.

This is principalization: after the succession the weak transform of J is the unit ideal, so the total transform of J is a product of powers of the exceptional divisors. Every centre lies in the locus of maximal order of the current weak transform. Hironaka writes the boundaries as \mathrm{red}(f_i^{-1}(E_i \cup B_i)), Main Theorem II as \mathrm{red}(f_i^{-1}(E_i) \cup f_i^{-1}(D_i)): the same reduced subscheme. For J = \mathcal{O}_X the empty succession satisfies every condition.

Conditions (1), (3) and the first half of (4) are those of Main Theorem II (Definition 4.2.1.1).

Let X be a smooth algebraic k-scheme, k of characteristic zero, and J_1, \dots, J_e (e \ge 1) ideal sheaves on X with nonzero stalks. Then there is a finite succession of monoidal transformations f_i \colon X_{i+1} \to X_i with non-singular irreducible centres D_i \subseteq X_i such that

1. every weak transform J_j(i) of every J_j has a constant positive order along D_i;

2. at every point of the end result X_r at least one of the weak transforms J_j(r) is the unit ideal.

This is Hironaka's trivialization of a system of coherent sheaves of ideals: after the succession the common cosupport of the weak transforms is empty. The constant of (1) may depend on i and j. The condition on the centres is HasRegularIrreducibleCenters (Definition 4.2.1.1), and condition (1) is HasConstantPositiveOrderAlongCenters for each J_j (Definition 4.2.3.1); the weak transforms are those of Main Theorem II (Definition 2.4.1.2).

There is a family BP of blow-up sequence functors on triples, one for each field k of characteristic zero, assigning a succession of blow-ups \Pi \colon X_r \to X to every triple (X, I, E) of a smooth equidimensional algebraic k-scheme, an ideal sheaf with nonzero stalks and a simple normal crossings divisor family, such that

1. every centre is smooth over k and has simple normal crossings with the total transform of E at its stage;

2. \Pi^* I is the ideal sheaf of a simple normal crossings divisor;

3. \Pi is an isomorphism over X \setminus (\operatorname{cosupp} I \cup \operatorname{Sing} E);

4. BP commutes with smooth morphisms h \colon Y \to X (the value on the pulled-back triple is h^* BP(X, I, E), with empty blow-ups deleted when h is not surjective) and with change of the base field;

5. when E = \emptyset, BP commutes with closed embeddings.

What distinguishes Kollár's theorem from Main Theorem II(N) (Theorem 4.2.2.1) is that the succession is chosen for all inputs at once, by one rule, and the rule is compatible with smooth morphisms and field extensions; this is what lets the resolution of an arbitrary scheme be glued from local pieces (Theorem 4.2.6.3).

Conditions (1)–(3) are Triple.IsPrincipalizedBy (Definition 4.2.4.2); condition (4) is CommutesWithSmoothMorphisms and CommutesWithChangeOfFields (Definition 4.2.4.3, Definition 4.2.4.4), and condition (5) CommutesWithClosedEmbeddingsOfEmptyBoundary (Definition 4.2.4.5). The inputs are triples (Definition 4.2.4.1), and the vocabulary is that of blow-up sequences and their functors (Definition 2.4.1.1, Definition 2.4.3.1, Definition 2.4.4.1), divisor families and simple normal crossings (Definition 2.4.2.1, Definition 2.4.2.2, Definition 2.4.2.4).

Let X be a smooth algebraic k-scheme, k of characteristic zero, and W a nowhere dense closed subscheme of X. Then there is a finite succession of monoidal transformations f_i \colon X_{i+1} \to X_i with non-singular centres D_i \subseteq X_i such that

1. the end result X_r is non-singular;

2. every centre lies over W: D_i \subseteq \bar f_i^{-1}(W), where \bar f_i = f_0 \circ \cdots \circ f_{i-1} \colon X_i \to X;

3. the associated reduced scheme \mathrm{red}(\bar f_r^{-1}(W)) of \bar f_r^{-1}(W) is defined by an invertible sheaf of ideals on X_r and has only normal crossings.

This is the simplification of an algebraic boundary: after blowing up over W only, the inverse image of W becomes a normal crossings divisor. Hironaka asks nothing of W beyond nowhere density; W may be empty, for which the empty succession satisfies every condition.

The statement uses regularity (Definition 2.1.2.1), invertible ideal sheaves (Definition 2.1.4.1) and Hironaka's normal crossings (Definition 2.1.5.1).