The coordinates of Kollár's proof of Theorem 92 at a point of the cosupport #
The proof of Theorem 92 ([Kol07, 95]) opens by choosing coordinates at a point p of the
cosupport: local sections x₁, x₁' ∈ MC(I) with H = (x₁ = 0) and H' = (x₁' = 0), further
coordinates x₂, …, x_{s+1} with Eⁱ = (x_{i+1} = 0) for the components of E, and general
x_{s+2}, …, xₙ, such that both x₁, x₂, …, xₙ and x₁', x₂, …, xₙ are local coordinate systems.
This file proves that statement (exists_coordinates).
The argument. At p ∈ cosupp(I, m) the hypersurface of maximal contact H passes through
p (H ⊆ MC(I) as ideal sheaves and p ∈ V(MC(I)), coe_support_MC), so the simple normal
crossings of H + E at p ([Kol07, Definition 24]) give a regular system of parameters z of
𝒪_{X,p} with H_p = (z_{p₀}) and Eⁱ_p = (z_{c(i)}) for the components through p, c
injective: Kollár's x₁ = z_{p₀} and x_{i+1} = z_{c(i)}. Likewise H' + E gives z' with
H'_p = (z'_{p₀'}), Eⁱ_p = (z'_{c'(i)}). The two equations of Eⁱ_p differ by a unit, so the
classes ē_i and ē'_i in 𝔪/𝔪² span the same subspace U, and the class of x₁' = z'_{p₀'}
is not in U (it is a basis vector of the second basis outside the c'-indices). The ring-level
lemma exists_cons_regularSystem_common then produces y containing the Eⁱ-coordinates such
that (x₁, y) and (x₁', y) are both regular systems of parameters; x₁, x₁' ∈ MC(I)_p because
H_p, H'_p ⊆ MC(I)_p.
A hypersurface of maximal contact (static form) passes through every point of cosupp(I, m):
H ⊆ MC(I) and p ∈ V(MC(I)) ([Kol07, Definition 79]).
Two generators of one principal ideal of a domain have classes spanning the same line of the cotangent space.
The coordinates of Kollár's proof of Theorem 92 ([Kol07, 95]): at a point p of
cosupp(I, m), local sections x₁, x₁' ∈ MC(I)_p with H_p = (x₁), H'_p = (x₁'), and
y = (x₂, …, xₙ) such that (x₁, y) and (x₁', y) are both regular systems of parameters and
every component Eⁱ of E through p is (y_{c(i)} = 0), c injective.