The tuned ideal I + D(I)²: general properties, and Kollár's Examples 11 and 106 #
For an ideal I of a ℚ-algebra, the tuned ideal I + D(I)² has the same derivative as I,
D(I + D(I)²) = D(I), by monotonicity of D and the Leibniz rule; consequently
D(I + D(I)²)² = D(I)² ⊆ I + D(I)², which says that I + D(I)² is D-balanced with respect to 2
and MC-invariant (the case s = 2 of [Kol07, Proposition 99, (5) and (8)]). The three theorems of
this file state these facts in the form the examples use; they are the theorems
Ideal.derivative_sup_derivative_sq, Ideal.isDBalanced_sup_derivative_sq and
Ideal.mcInvariant_sup_derivative_sq of HironakaExamples/Balanced/Tuned.lean.
The worked instances are HironakaExamples/Balanced/Example11.lean and Example106.lean, imported
here. In ℚ[x, y, z] the partial derivatives of x² + y³ − z⁶ are 2x, 3y², −6z⁵, so
D(I) = (x, y², z⁵), and I + D(I)² is generated by seven elements (z¹⁰ = z⁴(z⁶ − y³) + y³z⁴);
x² ∈ D(I)² but x² ∉ I (evaluate at (1, 0, 1)), so I itself is neither D-balanced nor
MC-invariant; both I and W(I) lie in 𝔪₀² and contain an element outside 𝔪₀³ (two partial
derivatives and the evaluation at the origin); the common zeros of x, y², z⁵ form the origin; and
the two restrictions x ↦ 0, x ↦ −y² send the seven generators into (y⁴, y²z⁵, y³z⁴, z⁶ − y³)
and hit its generators ([Kol07, Example 11]). In ℚ[x, y, z, w],
D(x³ − y², x⁴ + xz² − w³) = (x², y, z², xz, w²), y² ∈ D(I)² but y² ∉ I (evaluate at
(1, 1, 0, 1)), and the orders and the cosupport follow the same pattern ([Kol07, Example 106]).
D(I + D(I)²) = D(I): the form of Ideal.derivative_sup_derivative_sq used by the examples.
I + D(I)² is D-balanced with respect to 2 (Ideal.isDBalanced_sup_derivative_sq).
I + D(I)² is MC-invariant with respect to 2: D(I + D(I)²)² ⊆ I + D(I)²
(Ideal.mcInvariant_sup_derivative_sq).