Monomials of an snc family are ideal sheaves of snc divisors #
[Kol07, Theorem 35 (2)] says that Π^* I is "the ideal sheaf of a simple normal crossing divisor";
the predicate IsIdealOfSncDivisor asks for an snc family F and exponents a with
I = ∏_j (F^j)^{a_j}. The fine monomial E.monomial a of
Hironaka.Resolution.Algebraic.MarkedOrderReduction.MonomialPart ([Kol07, Definition–Lemma 110]
read on the irreducible components) carries one exponent per irreducible component of the members of
E; this module realises it as an IsIdealOfSncDivisor through the component family of E —
the family of all irreducible components of all members, indexed by (member, generic point) in the
lexicographic order — which is snc when E is ([Kol07, Definition 24]: "each E^i is smooth"; a
member may have several irreducible components, [Kol07, Notation 64 (3)]: "Each E^i is allowed to
be reducible or empty", and the sentence after [Kol07, Definition–Lemma 110]): a component has the
stalk of its member at each of its points (stalkIdeal_componentFamily_eq of
Hironaka.Resolution.Algebraic.Kol07.ComponentwiseBlowUp), two components of one member never meet
(eq_of_mem_support_componentFamily), so the coordinates of IsSncAt are those of E. This is how
the principalization theorem's clause (2) is verified for the monomial ideal left at the end of
Kollár's order reduction.
DivisorFamily.components,components_component,support_components_component_le;DivisorFamily.monomial_eq_prod_components:E.monomial a = ∏_p (component p)^{a p};Snc.isSnc_components,Snc.isIdealOfSncDivisor_monomial;IsIdealOfSncDivisor.comap_of_smooth: the property pulls back along smooth morphisms (isSnc_comap_of_smoothofHironaka.Scheme.BlowUpSequence.PullbackSnc;comapis multiplicative).
Sources: [Kol07, Theorem 35 (2); Definition 24; Notation 64 (3); Definition–Lemma 110]; [BM08, (5.2)].
The component family of E — the irreducible components of all the members, indexed by the
member and the generic point of the component (the componentFamily of the member), in the
lexicographic order (the members may be reducible, [Kol07, Notation 64 (3)]; the sentence after
[Kol07, Definition–Lemma 110] speaks of the "irreducible components of some of the E^i").
Equations
- One or more equations did not get rendered due to their size.
Instances For
A component lies in its member.
The fine monomial 𝒪_X(−∑_D a_D D) is the product over the component family of the powers of
the components.
[Kol07, Definition 24] for the components of members that may be reducible
([Kol07, Notation 64 (3)]): the component family of a simple normal crossing family is simple normal
crossing — a component is regular with the stalks of its member, and at a point the components
through it are in bijection with the members through it (two components of one member are disjoint),
so Kollár's coordinates for E serve.
[Kol07, Theorem 35 (2)] in the vocabulary of IsIdealOfSncDivisor: a monomial of a simple
normal crossing family (one exponent per irreducible component) is the ideal sheaf of a simple
normal crossing divisor in the sense of that predicate — of the component family, with the same
exponents.
[Kol07, 34.1] for clause (2) of Theorem 35: the ideal sheaf of an effective divisor supported on
an snc family pulls back along a smooth morphism to one — the family pulls back to an snc family
(isSnc_comap_of_smooth) and comap is multiplicative (comap_finset_prod, comap_pow).