Tuning of ideals, II: Kollár's Corollary 101 #
Kollár's Corollary 101 [Kol07, Corollary 101] (Tuning of ideals, II): for m = max-ord I ≥ 1 and
s = r · lcm(2, …, m) with r ≥ m − 1, the tuned ideal W_s(I) is MC-invariant and D-balanced,
and a smooth blow-up sequence is of order ≥ m starting with (X, I, m, E) iff it is of order
≥ s starting with (X, W_s(I), s, E). "Everything follows from (99) and (100), except for the
role played by E."
- The properties of
W_s(I)([Kol07, Proposition 99 (5) and (8)]; the maximal order is [Kol07, Theorem 54.2 (i)] for the tuned ideal). They are inclusions and orders of ideal sheaves, checked at stalks. An inclusion of ideal sheaves on a Jacobson scheme (every scheme locally of finite type over a field,LocallyOfFiniteType.jacobsonSpace) holds as soon as it holds at the closed points (le_of_forall_stalkIdeal_le_of_isClosed); at a closed point the residue field is algebraic overk(isAlgebraic_residueField_of_isClosed), so the stalk carriesk-linear coordinates whose derivations span thek-derivations (exists_regularCoords_stalk_of_isAlgebraic), and the stalk bridgesstalkIdeal_W,stalkIdeal_MC,stalkIdeal_derivative,stalkIdeal_derivativeIterwithderivative_eq_D,derivativeIter_eq_Dpowturn the sheaf statements into the statements in coordinates ofHironaka.Local(RegularCoords.isMCInvariant_W,RegularCoords.isDBalanced_W,RegularCoords.ord_W,RegularCoords.W_eq_top_of_ord_lt), whose hypothesisord_p I ≤ mism = max-ord I. Formax-ord W_s(I) = s: the order at any point is bounded by the order at a closed point it specializes to (ord_le_ord_of_specializes,exists_closedPoint_specializes), where it issor0; and a point of ordermexists becausemax-ord I = m ≥ 1is a supremum of naturals bounded bym(were every order≤ m − 1, so would be the supremum,maxOrd_le_iff), and it specializes to a closed point of orderm, whereW_s(I)has order exactlys. - The reduction to Theorem 100.
m ∣ sands ≥ 1gives = m s'withs' ≥ 1, andtuning_iff_WofHironaka.Resolution.Algebraic.Tuning.Assembleis the statement atJ = W_{ms'}(I). - The role of
E. A sequence of order≥ mfor(X, I, m, E)is one for the empty family (its centres have simple normal crossings with the exceptional divisors, a sub-family of the induced familiesE_i: theIsSubfamilyVialemmas ofHironaka.Snc) together with the snc clauses forE_i; conversely the two recombine. The clause is a condition on(Z_i, E_i)alone. - Corollary 101.
s = r · Lcm mis divisible bym(dvd_Lcm) and positive forr ≥ 1, so the reduction gives the equivalence; with the properties above, the corollary as printed, except for the added hypothesisr ≥ 1: Kollár'sr ≥ m − 1allowsr = 0whenm = 1, wheres = 0,W_0(I) = 𝒪_Xand the equivalence fails.
m = max-ord I bounds the order of every stalk (the hypothesis of the coordinate lemmas on
Proposition 99 at each point).
The tuned ideal is MC-invariant, D-balanced, of maximal order s #
[Kol07, Corollary 101; Proposition 99 (5)]: for m = max-ord I ≥ 1 and s ≥ 1, W_s(I) is
MC-invariant with respect to s, that is MC_s(W_s(I)) · D(W_s(I)) ⊆ W_s(I). Proved in coordinates
at the stalk of every closed point (RegularCoords.isMCInvariant_W), through the closed-point
reduction and the stalk bridges.
[Kol07, Corollary 101; Proposition 99 (8)]: for m = max-ord I ≥ 1 and r ≥ m − 1,
W_{r·lcm(2,…,m)}(I) is D-balanced with respect to r · lcm(2, …, m). Proved in coordinates at the
stalk of every closed point (RegularCoords.isDBalanced_W).
The order of the tuned ideal at a closed point: at most s, and exactly s where I has order
m (RegularCoords.ord_W; where ord I < m, W_s(I)_p = 𝒪_p by
RegularCoords.W_eq_top_of_ord_lt).
For m = max-ord I ≥ 1, max-ord W_s(I) = s (the analogue of [Kol07, Theorem 54.2 (i)] for
W_s). The bound at any point comes from a closed point it specializes to; the value is attained at
a closed specialization of a point of order m, which exists because a supremum of naturals bounded
by m and equal to m ≥ 1 is attained.
The role of E #
The role of E in the proof of [Kol07, Corollary 101]: a smooth blow-up sequence is of order
≥ m starting with (X, I, m, E) iff it is of order ≥ m starting with (X, I, m) (the empty
family) and every centre has simple normal crossings with the total transform E_i. The direction
→ drops E to the sub-family of exceptional divisors (IsSubfamilyVia), the direction ←
recombines.
The reduction to Theorem 100 and Corollary 101 #
The reduction of [Kol07, Corollary 101] to Theorem 100: for m ∣ s and s ≥ 1, Theorem 100 at
the upper bound J = W_{ms'}(I) (tuning_iff_W) with s = m s'.
Kollár's Corollary 101 [Kol07, Corollary 101], the equivalence for every divisor family:
s = r · lcm(2, …, m) is divisible by m (dvd_Lcm) and, under the additional hypothesis r ≥ 1,
positive, so the reduction to Theorem 100 applies. Kollár's hypothesis r ≥ m − 1 is part of his
statement and is kept so that the corollary reads as printed (the D-balanced clause of
corollary101 needs it, isDBalanced_W); the equivalence does not need it, because Theorem 100
asks only that s be positive and divisible by m. The hypothesis r ≥ 1 is not in Kollár's
statement: it excludes only m = 1, r = 0, where s = 0, W_0(I) = 𝒪_X and the printed
equivalence fails (every smooth blow-up sequence is of order ≥ 0 for (𝒪_X, 0)).
Kollár's Corollary 101 [Kol07, Corollary 101] (Tuning of ideals, II) as printed: W_s(I) is
MC-invariant and D-balanced with respect to s = r · lcm(2, …, m), and a smooth blow-up sequence is
of order ≥ m starting with (X, I, m, E) iff it is of order ≥ s starting with
(X, W_s(I), s, E) — with the additional hypothesis r ≥ 1, which excludes only m = 1, r = 0
(there s = 0 and W_0(I) = 𝒪_X, and the equivalence fails: every smooth blow-up sequence is of
order ≥ 0 for (𝒪_X, 0)). Kollár's hypothesis "E snc" is part of his statement — a triple of
[Kol07, Notation 64] — and is kept so that the corollary reads as printed; the proof does not need
it, because tuning changes only the ideal: the three clauses hold for every divisor family, the
order-≥ m condition involving E only through the snc clauses on the centres, which are the same
on both sides of the equivalence.