The exceptional set of a blow-up along a smooth centre of codimension at least two #
For X smooth of relative dimension n over a field k and a closed subscheme Z smooth of
relative dimension n − r over k, 2 ≤ r ≤ n, the exceptional set of the monoidal
transformation Z.blowUpπ is exactly the exceptional divisor |F|: the single-blow-up case of
Kollár's remark that for nontrivial blow-ups the exceptional set is the total exceptional divisor
[Kol07, Definition 25]. The two inclusions are exceptionalSet_blowUpπ_subset_support
(Hironaka.Scheme.BlowUp.ExceptionalSetSupport; the blow-up map is an isomorphism off the centre)
and support_comap_subset_exceptionalSet (Hironaka.Scheme.BlowUp.ExceptionalSetSmooth) applied
with the smoothness of F → Z of relative dimension r − 1,
smoothOfRelativeDimension_exceptionalMap of Hironaka.Scheme.Smooth.ExceptionalDivisorSmooth.
The mathematics. Ex(π) ⊆ |F| because π is an isomorphism over X ∖ Z. For |F| ⊆ Ex(π),
let x' ∈ F and suppose π is an open immersion on an open U' ∋ x'. Then F ∩ U' → Z, the
base change of π|_{U'} along Z → X, is an open immersion, hence smooth of relative dimension
0; it is also the restriction of F → Z to the open F ∩ U' of F, hence smooth of relative
dimension r − 1; its source is nonempty; and a morphism with nonempty source has one relative
dimension (SmoothOfRelativeDimension.eq_of_nonempty, from the rank of the Kähler differentials
on an affine piece). So r − 1 = 0, contradicting r ≥ 2.
The exceptional set of the blow-up along a smooth centre of codimension r ≥ 2 is the
exceptional divisor [Kol07, Definition 25]: for X smooth of relative dimension n over k and
Z ⊆ X smooth of relative dimension n − r over k with 2 ≤ r ≤ n,
Ex(Z.blowUpπ) = |Z.exceptionalDivisor|.