Transforms of ideal sheaves under a blow-up #
For the blow-up π : W' → W along Z with exceptional divisor E, an ideal I on W has the
total transform I* = I·𝒪_{W'} [Hau14, Definition 6.1], the strict transform
Iˢ = ⋃_{i ≥ 0} (I* : I_E^i), the union of colon ideals [Hau14, Definition 6.2], and, for c at
most the order of I along Z, the controlled transform I^!, the unique ideal with
I* = I_E^c · I^! [Hau14, Definition 6.8], whose case c = ord_Z I is the weak transform I^g.
Hironaka defines the weak transform under a monoidal transformation with non-singular irreducible
centre D as the quotient of f⁻¹(J) by the m-th power of the ideal of f⁻¹(D), m = ν(J_x) at
the generic point x of D, noting that m = ν(f⁻¹(J)_y) at the generic point y of f⁻¹(D)
[Hir64, Ch. 0, §5, p. 142]; so m is the largest power of that ideal dividing f⁻¹(J), the
exponent used here. The strict transform of a closed subscheme is the subscheme cut out by the
sections supported on the exceptional divisor, i.e. the I_E-power torsion [TSPA, Tag 080D].
The transforms are defined along any morphism π : B ⟶ X with a distinguished ideal sheaf E on
B, from the colon and the saturation of ideal sheaves: strictTransformAlong π E J,
controlledTransformAlong π E J c, exceptionalOrderAlong π E J (the supremum in ℕ of the
exponents c with E ^ c ∣ J.comap π, 0 when unbounded, i.e. when J.comap π = 0) and
weakTransformAlong π E J. Along the blow-up D.blowUpπ with E = D.exceptionalDivisor they are
the transforms of the main theorems: Hironaka's weak transform J.weakTransform D (with the
convention for a disconnected centre stated on its docstring), the strict transform
Y.strictTransform D of a closed subscheme, and Hironaka's red(f⁻¹(E) ∪ f⁻¹(D)),
E.reducedTransform D.
Boundary cases: J.comap π = 0 gives exceptional order 0 and weak transform 0; c = 0 gives
the total transform as controlled transform.
The strict transform of J along π with respect to the divisor E: the E-power torsion
saturation of the total transform J.comap π [Hau14, Definition 6.2]; [TSPA, Tag 080D]. The
strict transform of a closed subscheme of X under the blow-up is this along D.blowUpπ with
E the exceptional ideal.
Equations
- J.strictTransformAlong π E = (J.comap π).saturate E
Instances For
The controlled transform of J with control c, (J* : E^c) [Hau14, Definition 6.8];
[Kol07, Definition 60, (60.1)]; for E ^ c ∣ J* it is the unique ideal sheaf with
J* = E^c · J^! (pow_mul_colon_of_dvd, colon_pow_eq_of_mul_eq).
Equations
- J.controlledTransformAlong π E c = (J.comap π).colon (E ^ c)
Instances For
The exceptional order: the largest m such that E ^ m divides J.comap π, as a natural
number (0 if unbounded, i.e. J* = 0). For a non-singular irreducible centre in a non-singular
scheme this is ν(J) at the generic point of the centre [Hir64, Ch. 0, §5, p. 142].
Instances For
The weak transform, the controlled transform with the exceptional order as control
[Hir64, Ch. 0, §5, p. 142]; [Hau14, Definition 6.8] (I^g = I_E^{-ord_Z I} · I*).
Equations
- J.weakTransformAlong π E = J.controlledTransformAlong π E (AlgebraicGeometry.exceptionalOrderAlong π E J)
Instances For
Hironaka's weak transform of J by the monoidal transformation with centre D
[Hir64, Ch. 0, §5, p. 142; Ch. I, §3, p. 176]: for D non-singular irreducible and
m = ν(J_x) at the generic point x of D, f⁻¹(J) is divisible by the m-th power of the
ideal of f⁻¹(D), and the weak transform is the quotient.
Relation to the source.
- Interpretation. For a disconnected centre
Dthe weak transform divides the pull-back ofJby a single power of the whole exceptional divisor:AlgebraicGeometry.Scheme.IdealSheafData.weakTransformAlongis the colon ideal(J.comap π).colon (E ^ c)withc = exceptionalOrderAlong π E J = sSup {c | E ^ c ∣ J.comap π}for the whole exceptional divisorE, that is, the least of the orders along the components. Kollár's componentwise division [Kol07, 58 and Definition 60], stated under $\operatorname{ord}_{D_k} J = d$ for every component $D_k$, agrees with it only under equimultiplicity (the same order along every component ofD), which holds wherever this definition enters a main theorem: Main Theorems II and II(N) and Corollary 1 require every centre to be non-singular and irreducible, II(N) and Corollary 1 moreover of constant order on the centre, and the centres of the resolution algorithm lie in the maximal-order locus.
Equations
Instances For
The strict (birational) transform of the closed subscheme Y under the blow-up with centre
D [Hir64, Ch. 0, §2, pp. 129–130]; [Kol07, Definition 30, 30.2]:
AlgebraicGeometry.Scheme.IdealSheafData.strictTransformAlong.
Equations
Instances For
Hironaka's red(f⁻¹(E) ∪ f⁻¹(D)) [Hir64, Main Theorem II (iii); Main Theorem II(N) (3)]:
the reduced closed subscheme of the blow-up whose set of points is f⁻¹(E) ∪ f⁻¹(D), as the
vanishing ideal sheaf of that closed set (II(N) writes red(f⁻¹(E ∪ D)), the same reduced
subscheme: both are the reduced structure on f⁻¹(|E| ∪ |D|)).