The total transform in the chart coordinates #
Hauser's computation (the proof of [Hau14, Proposition 5.3]; [Hau14, Proposition 5.4 (6)–(7)];
[Hau03, Appendix C]): on the chart of x_j of the blow-up of Z = (x_1 = ⋯ = x_r = 0), the chart
functions are y_i = x_i/x_j for i < r, i ≠ j, and y_i = x_i otherwise
[Kol07, Definition 60]; the exceptional divisor is (y_j), a component (x_c = 0) containing the
centre has total transform (x_c) = (y_c y_j) — or (y_j) when c = j — and a transversal
component (x_c = 0) has total transform (y_c). At a point p of the chart the chart functions
are étale coordinates (Hironaka.Scheme.Smooth.ChartEtale), so by
Hironaka.Scheme.Snc.EtaleParameters the local ring is regular and the chart functions vanishing at
p are part of a regular system of parameters; a chart function not vanishing at p is a unit
there. Reading the total transforms through this regular system of parameters gives exactly the four
shapes of TotalTransformData.
Why the lemma holds. In the chart ring Γ(U)[J/x_j] one has x_i = y_i · x_j for i < r
(algebraMap_eq_ratio_mul) and x_j = y_j; the ideal J Γ(U)[J/x_j] is (x_j). Localizing at
p, a coordinate y_c ∈ p becomes the parameter z_{σ c}, a coordinate y_c ∉ p a unit:
(y_c y_j) is (z_{σ c} z_{σ j}) or (z_{σ j}), (y_c) is (z_{σ c}) or (1). The indices
σ c are distinct and differ from σ j for c ≠ j.
The chart functions y_i = x_i/x_j (i < r, i ≠ j) and y_i = x_i (otherwise) of the chart
of x_j [Kol07, Definition 60], as in etale_aeval_chartCoordinates.
Equations
- One or more equations did not get rendered due to their size.
Instances For
x_i = y_i y_j in the chart of x_j, for i < r, i ≠ j [Kol07, Definition 60].
The chart functions are étale coordinates of the chart ring over k (the k-algebra structures
through f).
The chart computation (the proof of [Hau14, Proposition 5.3]; [Hau14, Proposition 5.4 (6)–(7)];
[Hau03, Appendix C]): at a prime q of the chart ring of x_j lying on the exceptional divisor
(y_j ∈ q), for a family of ideal sheaves D i of X whose members through x₀ are the
coordinate hyperplanes (x_{c' i} = 0) on the chart (c' injective) and whose other members miss
the chart, the exceptional ideal (x_j) and the total transforms (x_{c' i}) (and (1)), extended
to the local ring at q, have the shape TotalTransformData: the local ring is regular with a
regular system of parameters containing the vanishing chart functions, F = (y_j), and each total
transform is (y_c y_j), (y_c), (y_j) or (1).