Kollár's chart formula and the map from the chart ring to the germ ring #
In a blow-up chart Φ of index i over an adapted chart φ, a section f composed with the
blowing-up reads, in the chart coordinates, as f ∘ φ⁻¹ ∘ ψ⁻¹ composed with the substitution
blowUpChartMap σ i: Kollár's f(y_1 y_r, …, y_{r−1} y_r, y_r, …, y_n) Kol07, Definition 60,
(60.2)–(60.3), Bierstone–Milman's formal substitution in the Taylor expansion BM97, §3,
"Blowing-up" (totalTransform_rep, from IsBlowUpChart.comm). When f ∈ I_Y^m along the
centre, the transform u_i^{-m} f(…) is an analytic function on the chart (exists_rep_div: the
pulled-back germs lie in (u_i^m) at the points of the exceptional divisor, so the iterated
canonical quotient of Hironaka.Manifold.BlowUp.Transform.Division is analytic and satisfies
u_i^m · g = f ∘ π). Two blow-up charts at a point give representatives differing by a unit
(rep_div_unit: both scaling coordinates generate the stalk of I_F, hence are associated in the
domain stalk), as Kollár remarks after (60.3). The chart ring R' = R[x_j / x_r : j < r] of the
algebraic strand (IsLocalRing.chartRing, for the centred adapted coordinates x in Kollár's
order: the scaling coordinate at position r, the other centre coordinates before it) maps to
the germ ring at a point a' of the chart over the centre: the stalk map
germMap π : 𝒪_{M,a} → 𝒪_{M',a'} sends x_r to the scaling coordinate u_i and x_j (j < r)
to u_i u_j, so it extends by IsLocalization.lift to R[1/x_r] with values in the fraction
field of the domain 𝒪_{M',a'}, and on the subalgebra R' the values are the germs u_j and
germMap π (·) themselves, hence lie in 𝒪_{M',a'} (exists_chartRing_hom, with
χ ∘ algebraMap = germMap π, x_j / x_r ↦ u_j, x_r ↦ u_i). Through any such χ, the stalk of
the birational transform π_*^{-1}(I, m) is the image of the algebraic transform
transformIdeal x r I_a m (birationalTransform_stalkIdeal_eq_map_transformIdeal): the colon
(π⁻¹(I)_{a'} : u_i^m) is spanned by the u_i^{-m}(f_j ∘ π) = χ(f_j / x_r^m) for local
generators f_j ∈ I_a ⊆ (x_1, …, x_r)^m (chartCenter_eq_stalkIdeal_idealSheaf,
exists_algebraMap_pow_mul_eq), and conversely every generator of the algebraic transform maps
into the colon.
This identification connects the analytic birational transform with the algebraic one of
Hironaka.Algebra.Local.BirationalTransform; it is used for the derivative rule of
Hironaka.Manifold.BlowUp.Transform.DerivChart.
f ∘ π at a point of a
blow-up chart is f ∘ φ⁻¹ ∘ ψ⁻¹ composed with the substitution blowUpChartMap σ i
[Kol07, Definition 60, (60.2)–(60.3)].
For f ∈ I_Y^m along Y, the transform
u_i^{-m} · f(y_1 y_r, …) is an analytic function g on the blow-up chart with
u_i^m · g = f ∘ π ∘ Φ⁻¹ [Kol07, Definition 60, (60.3)].
The representatives of the transform of a germ in two blow-up charts at a point differ by a unit (the remark after (60.3) in [Kol07, Definition 60]).
At a point of the centre, the centred adapted coordinates in Kollár's order are
the adapted coordinates, and the chart centre chartCenter x r of the algebraic strand is the
stalk of I_Y.
The chart ring IsLocalRing.chartRing maps to the germ ring at a point of the chart over
the centre, with χ ∘ algebraMap = germMap π, x_j / x_r ↦ u_j (j < r) and x_r ↦ u_i.
Through any chart-ring map χ compatible with the
germ map, the stalk of the marked transform of (I, m) is the image of the algebraic transform
transformIdeal x r I_a m.