The total transform of a simple normal crossings divisor is snc #
Kollár: if the centre has simple normal crossings with E, then
Π⁻¹_tot(E) := Π⁻¹_*(E) + Ex_tot(Π) "is a simple normal crossing divisor, called the total
transform of E" [Kol07, Definition 25]; Bierstone–Milman: the admissibility condition
"guarantees that E_{i+1} is a collection of smooth hypersurfaces having only normal crossings"
[BM97, (1.2)]. Over a point a of the centre, a chart adapted to Y that is an snc chart of F
at a lifts to blow-up charts of every index i ([BM88, Definition 4.1 (2)]); in the blow-up
chart of index i the exceptional divisor is {u_i = 0} and the strict transform of the component
{z_{c j} = 0} is {u_{c j} = 0} when c j ≠ σ i (both a block coordinate u_k, k ≠ i, and a
coordinate off the block) and misses the chart when c j = σ i — so the adapted coordinates of the
total transform are (u_i, (u_k)_{k ≠ i}, w), with the indices c on the strict transforms and
σ i on the exceptional divisor (totalTransformIdx). Off the exceptional divisor the blowing-up
is a local isomorphism ([BM88, Definition 4.1 (1)]), the strict transforms coincide with the
preimages there, and the snc chart of F at π p transports along a local inverse
(transportChart). Each strict transform is a closed smooth hypersurface (from the charts over
H ∩ Y, isClosedSubmanifold_strictTransform_of_charts), and the family stays locally finite
because each strict transform lies in the preimage of its component. This is the boundary step of
every blow-up of the resolution algorithm (Hironaka's E_{λ+1} = red(f⁻¹(E_λ) ∪ f⁻¹(D_λ)),
[Hir64, Main Theorem II'(N) (iii), p. 156]).
The coordinate indices of the total transform at a point p over a, from the indices cidx
of an snc chart at a and the block coordinate σ i of the blow-up chart: the strict transform of
a component through p (which lies over a component through a) keeps its index, the exceptional
divisor gets σ i.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A point of the strict transform of a component lies over the component.
Off the exceptional divisor the strict transform of a component is its preimage.
The index of a strict-transform component of the total transform is the index of its component.
The index of the exceptional component of the total transform in the blow-up chart of index
i is σ i.
The adapted coordinates (u_i, (u_k)_{k ≠ i}, w) of the total transform: over a ∈ Y, a
blow-up chart Φ of index i over a chart φ adapted to Y which is an snc chart of F at a
is an snc chart of the total transform at every point of its source over a, with the indices
cidx on the strict transforms and σ i on the exceptional divisor.
Off the exceptional divisor, the snc chart of F at π p transported along a local inverse
of π is an snc chart of the total transform at p.
If Y has simple normal crossings with the snc divisor F, the total transform of F under
the blowing-up along Y is a simple normal crossings divisor ([Kol07, Definition 25];
[BM97, (1.2)]).