Admissible chain strata #
The hypothesis of the one-blow-up statement CP4 of the embedded desingularization
(Hironaka.Resolution.Algebraic.Wlo05.EmbeddedChainRelative,
Hironaka.Resolution.Algebraic.Wlo05.EmbeddedCP4) and the conclusion of CP3. A centre Z = (f₀ = ⋯ = f_{l−1} = 0, e_j = 0 (j ∈ s)), a chain stratum at p through the protected component Γ, is
admissible when the exponent of the stratum's OWN level along the members s is positive (the
condition (★) below): ∃ k ∈ s, b k ≠ 0 at level l = 0, ∃ k ∈ s, a (l − 1) k ≠ 0 at level l ≥ 1. The weaker condition K ≤ Z (the centre lies in V(K)) is the CUMULATIVE condition M⁰ M₀ ⋯ M_{l−1} ∈ (e_s) (span_mul_chainKIdeal_le_stratum_iff in
Hironaka.Resolution.Algebraic.Wlo05.EmbeddedCP4Algebra) and does NOT suffice: for K = e·(f₁, e′)
along Γ = V(f₁, g) and Z = V(f₁, e), K ≤ Z holds but the mark-1 transform in the chart e = ε, f₁ = ε f₁′ is (ε f₁′, e′), which is no K-shape at the points of Γ′ ∩ F. CP3 derives (★) for
every centre of a run on a protected state
(Hironaka.Resolution.Algebraic.Wlo05.EmbeddedCP3Predicates); CP4 consumes it.
A third predicate covers the un-isolated run: the terminal-normal strata
Z = V(f₀, …, f_r, e_j : j ∈ s), s ≠ ∅ — ALL of Γ's equations plus boundary equations, so
Z ⊆ Γ locally without containing Γ — arise as the pushed-forward centres of a boundary pass on
an un-isolated state (in 𝔸² with coordinates g, e, boundary V(e) and ideal (g), the pass
over V(e) produces the centre V(g, e)); their transform needs no (★) and consumes no unit. They
never occur in a run on an isolated (protected) state, whose centres never contain all of Γ's
equations at once.
The predicates package the stratum with the chain data in ONE existential: the stratum, the
K-shape (or the un-isolated chain form) and the exponent condition are read in the SAME chain
coordinates. Conventions: the chain index i is the level i + 1; the top-level exponents b
are the level 0 (M⁰); the stratum's level is l ≤ r, so the last chain equation g = f_r is
never among the stratum's equations (Γ ⊄ Z locally). These statements are not in the literature.
Z is an admissible chain stratum for the K-shape K along Γ at p (the conclusion of
CP3 and the hypothesis of CP4, isolated form) — in chain coordinates shared with the K-shape of
K, Z_p = (f₀, …, f_{l−1}, z_s) with l ≤ r and s among the members' coordinates, and (★):
the stratum's own level carries an exponent along s (b at level 0, a (l − 1) at level
l ≥ 1).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Z is an admissible chain stratum for the un-isolated chain form of I along Γ at p
(the hypothesis of CP4 for the un-isolated ideal) — the same data with chainIdeal (the last
generator M⁰ M₀ ⋯ M_{r−1} g) in place of chainKIdeal.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Z is a terminal-normal stratum for the un-isolated chain form of I along Γ at p —
in chain coordinates shared with the chain form of I, Z_p = (f₀, …, f_r, z_s) with s a
NONEMPTY set of the members' coordinates: all of Γ's equations plus boundary equations, no
exponent condition.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A terminal-normal stratum contains Γ locally (Γ_p ≤ Z_p), and I has the chain form at
p.
An admissible stratum for the K-shape is in particular a chain-relative stratum
(IsChainStratumAt), and K has the K-shape at p.
An admissible stratum for the un-isolated form is a chain-relative stratum, and I has the
chain form at p.