Deleting the irreducible components of a regular scheme by monoidal transformations #
The degenerate case of Hironaka's Main Theorem II [Hir64, Main Theorem II, pp. 142–143]. When the
ideal sheaf J has order 0 at every point (d = 0, that is J = 𝒪_X), clause (iv), "J_r has
order < 0 at every point of X_r", holds exactly when X_r is empty, and a finite succession of
monoidal transformations with non-singular irreducible centres emptying X exists: blow up the
irreducible components of X one after another. On a regular Noetherian scheme the irreducible
components are pairwise disjoint (a regular local ring is a domain, so a point lies on one
component only), hence open. The blow-up of X along the reduced ideal of a component C is, over
C, the blow-up along the zero ideal, which is empty (blowUp.isEmpty_bot), and over the
complement of C an isomorphism, so its blow-up map is the open immersion of the union of the
other components. After one blow-up per component the last stage is empty, and every clause of
Main Theorem II holds: (i) each centre is an irreducible component of a regular scheme; (ii)
0 ≤ ν is trivial; (iii) the boundary E_i at each stage is the restriction of E₀ to the open
subscheme X_i ⊆ X — the reduced transform along an open immersion missing the centre is the
inverse image — and Hironaka's Definition 2 is a condition on stalks, which the stalk isomorphisms
of an open immersion preserve, while the centre, whose stalk ideals vanish on its support, has only
normal crossings with any such boundary; (iv) holds vacuously on the empty last stage.
exists_blowUpSequence_isEmpty_last_of_isRegular: the succession, for a regular Noetherian schemeXand a reduced closed subschemeE₀with only normal crossings;exists_blowUpSequence_isEmpty_last_of_smoothis the same for a scheme smooth and of finite type over a field of characteristic zero, the hypotheses of Main Theorem II.- The induction (
exists_blowUpSequence_isEmpty_last_aux) runs over the open immersionsu : Y ⟶ Xinto the fixedX, on the number of irreducible components ofXmeeting the range ofu, with boundaryE₀.comap uonY: the centre is the inverse image of the reduced ideal of a component through a point of the range, its blow-up map is an open immersion missing the centre (isOpenImmersion_blowUpπ_of_comap_eq_bot), and the next stage is the composite withu. Every property of a stage — regularity, the integrality of the centre, the reducedness and the normal crossings of the boundary — is read offXthroughuinstead of being established anew on the blow-up. - The transport lemmas:
IsSncBoundaryWith.comap_of_isOpenImmersion(Definition 2 along an open immersion),IsRegular.of_isOpenImmersion,isRegular_subscheme_of_forall_stalkIdeal_eq_bot,reducedTransform_eq_comap_of_forall_notMem_support, and the disjointness of the irreducible components of a regular scheme (hasDisjointIntegralComponents_of_isRegular).
Transport along an open immersion #
Regularity descends along an open immersion: the stalks of the source are stalks of the target.
Hironaka's Definition 2 [Hir64, Ch. 0, §5, Definition 2] transports along an open immersion
u: the regular system of parameters at u y, the minimal primes of the stalk of E and the
generators of the stalk of D are carried to y by the stalk isomorphism
(stalkEquivOfIsOpenImmersion), so u⁻¹ E has only normal crossings with u⁻¹ D when E has
only normal crossings with D.
IsSncBoundaryWith.comap_of_isOpenImmersion at D = X: normal crossings transport
along an open immersion.
Stalks along an open immersion #
The stalk of the inverse image of J along an open immersion u is zero iff the stalk of
J at the image point is zero: the stalk map is an isomorphism (stalkIdeal_comap).
A closed subscheme D whose stalk ideals vanish on its support has only normal crossings with
a closed subscheme E having only normal crossings: Definition 2 at x ∈ D with the empty set of
parameters generating D_x = 0.
A closed subscheme V(D) whose stalk ideals vanish on its support is regular when the ambient
scheme is: its stalks are the stalks of the ambient scheme (stalkQuotientEquiv by the zero
ideal).
The blow-up along a centre that is the whole scheme over an open #
If the centre D restricts to the zero ideal sheaf on an open U and every point off U is
off the support of D, the blow-up map π : D.blowUp ⟶ X is an open immersion missing the
support of D: over U the blow-up is the blow-up of the zero ideal sheaf, which is empty
(blowUp.restrictIso, blowUp.isEmpty_bot), so π lands in the complement V of the support,
over which it is an isomorphism [TSPA, Tag 02OS] (blowUp.isIso_π_restrict_compl_support); hence
π⁻¹(V) is all of D.blowUp and π is the isomorphism π ∣_ V followed by the inclusion of
V.
Hironaka's reduced transform red(π⁻¹(E) ∪ π⁻¹(D)) [Hir64, Main Theorem II (iii)] under a
blow-up whose map π is an open immersion missing the centre D is the inverse image π⁻¹ E of
the reduced E: the preimage of the centre is empty, and the vanishing ideal sheaf of the support
of the reduced π⁻¹ E is π⁻¹ E itself.
The irreducible components of a regular scheme #
On a regular scheme two irreducible components through a common point coincide: their generic
points are maximal points of X specialising to the point, and a regular local ring has a single
minimal prime (eq_of_specializes_of_isRegular at the zero ideal sheaf).
A regular scheme is reduced with pairwise disjoint irreducible components: a member of the
class HasDisjointIntegralComponents.
The succession #
The induction behind exists_blowUpSequence_isEmpty_last_of_isRegular, over the open
immersions u : Y ⟶ X into the fixed regular Noetherian X, on the number n of irreducible
components of X meeting the range of u, with the boundary E₀.comap u on Y. If Y is
empty the empty succession does. Otherwise some component C of X meets the range; the centre
is the inverse image D of the reduced ideal of C, an integral closed subscheme (open in
V(C)) whose stalk ideals vanish on its support, so it is regular and has only normal crossings
with the boundary; its blow-up map π is an open immersion missing D
(isOpenImmersion_blowUpπ_of_comap_eq_bot), along which the boundary is carried to
E₀.comap (π ≫ u), and the succession continues with the one the induction hypothesis gives for
π ≫ u, whose range meets one component fewer.
Deleting the irreducible components of a regular Noetherian scheme X: for a reduced
closed subscheme E₀ of X having only normal crossings there is a finite succession of monoidal
transformations of X whose centres are regular and irreducible, whose boundaries
E_i = S.boundarySeq E₀ i have only normal crossings with the centres, and whose last stage X_r
is empty: blow up the irreducible components of X one after another, each blow-up deleting its
component. This is the degenerate case d = 0 of Hironaka's Main Theorem II
[Hir64, Main Theorem II, pp. 142–143], in which clause (ii) is trivial and clause (iv) is
vacuous.
exists_blowUpSequence_isEmpty_last_of_isRegular for an algebraic scheme smooth over a field
k of characteristic zero, the setting of Main Theorem II [Hir64, Main Theorem II, pp. 142–143]:
such a scheme is Noetherian (isNoetherian_of_field) and regular (isRegular_of_smooth).