| import Mathlib |
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| namespace OAI |
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| namespace StableCoordinate |
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| open MvPolynomial |
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|
| abbrev R (n : ℕ) := MvPolynomial (Fin n) ℂ |
|
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| noncomputable def Q : R 4 := |
| X 1 ^ 2 - X 3 ^ 2 + X 0 * X 2 |
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| noncomputable def f : R 4 := |
| X 0 - 2 * Q * (Q * (X 1 + X 3) + X 0 * X 3) |
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| def IsCoordinate {n : ℕ} (hn : 0 < n) (g : R n) : Prop := |
| ∃ φ : R n ≃ₐ[ℂ] R n, φ g = X ⟨0, hn⟩ |
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| noncomputable def stabilize : R 4 →ₐ[ℂ] R 5 := |
| rename (Fin.castSucc : Fin 4 → Fin 5) |
|
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| def IsOneStableCoordinate (g : R 4) : Prop := |
| ∃ φ : R 5 ≃ₐ[ℂ] R 5, φ (stabilize g) = X 0 |
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| noncomputable def fiberIdeal (t : ℂ) : Ideal (R 4) := |
| Ideal.span ({f - C t} : Set (R 4)) |
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| abbrev FiberRing (t : ℂ) := R 4 ⧸ fiberIdeal t |
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| def IsAffineThreeSpaceFiber (t : ℂ) : Prop := |
| Nonempty (FiberRing t ≃ₐ[ℂ] R 3) |
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| def IsRectifiableFiber (t : ℂ) : Prop := |
| ∃ φ : R 4 ≃ₐ[ℂ] R 4, |
| Ideal.map φ.toRingHom (fiberIdeal t) = Ideal.span ({X 0} : Set (R 4)) |
|
|
| def PolynomialPackage : Prop := |
| f.totalDegree = 5 ∧ |
| IsOneStableCoordinate f ∧ |
| ¬ IsCoordinate (by decide : 0 < 4) f |
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| def FiberPackage : Prop := |
| ∀ t : ℂ, IsAffineThreeSpaceFiber t ∧ ¬ IsRectifiableFiber t |
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| def PrincipalPackage : Prop := PolynomialPackage ∧ FiberPackage |
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| / |
| theorem principal_package : PrincipalPackage := by |
| sorry |
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|
| end StableCoordinate |
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| end OAI |
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|
