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21cfc82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.AlgebraicGeometry.StableCoordinate.Definitions
import OAI.AlgebraicGeometry.StableCoordinate.AffineGeometry
import Mathlib.Algebra.MvPolynomial.Division
namespace OAI
/-!
# Schemes of the complex hypersurface fibers
An algebraic isomorphism of fiber coordinate rings induces an isomorphism of
affine schemes over `Spec ℂ`. Nonrectifiability excludes every ambient scheme
automorphism over the complex base, for each complex parameter.
-/
noncomputable section
namespace StableCoordinate
namespace GeometricInterpretation
open CategoryTheory AlgebraicGeometry MvPolynomial
open scoped AlgebraicGeometry
/-- The complex affine four-space in its standard spectrum presentation. -/
abbrev Ambient := Spec (CommRingCat.of (R 4))
/-- The fiber as a scheme, with its full coordinate-ring structure. -/
abbrev Fiber (t : ℂ) := Spec (CommRingCat.of (FiberRing t))
/-- The actual closed immersion defined by `f - t`. -/
def fiberEmbedding (t : ℂ) : Fiber t ⟶ Ambient :=
AffineGeometry.quotientEmbedding (fiberIdeal t)
instance fiberEmbedding_isClosedImmersion (t : ℂ) :
IsClosedImmersion (fiberEmbedding t) :=
AffineGeometry.quotientEmbedding_isClosedImmersion (fiberIdeal t)
/-- Standard identification of the ambient spectrum with affine four-space. -/
def ambientAffineSpaceIso :
Ambient ≅ AffineSpace (Fin 4) (Spec (CommRingCat.of ℂ)) :=
AffineGeometry.affineSpaceIsoOfAlgEquiv (AlgEquiv.refl : R 4 ≃ₐ[ℂ] R 4)
/-- The coordinate hyperplane's defining ideal. -/
def hyperplaneIdeal : Ideal (R 4) := Ideal.span {X 0}
theorem hyperplaneIdeal_isPrime : hyperplaneIdeal.IsPrime := by
apply (Ideal.span_singleton_prime (X_ne_zero (0 : Fin 4))).mpr
exact X_prime
/-- The fiber algebra condition proves its defining ideal is prime; this is not
an additional hypothesis about the source polynomial. -/
theorem fiberIdeal_isPrime_of_affineFiber (t : ℂ) (h : IsAffineThreeSpaceFiber t) :
(fiberIdeal t).IsPrime := by
obtain ⟨e⟩ := h
let : IsDomain (FiberRing t) := Function.Injective.isDomain e e.injective
exact (Ideal.Quotient.isDomain_iff_prime (fiberIdeal t)).mp inferInstance
/-- Geometric rectifiability of the embedded image by an arbitrary ambient
scheme automorphism over the complex base scheme. -/
def IsSchemeRectifiableFiber (t : ℂ) : Prop :=
∃ e : Ambient ≅ Ambient,
(e.hom ≫ Spec.map (CommRingCat.ofHom (algebraMap ℂ (R 4))) =
Spec.map (CommRingCat.ofHom (algebraMap ℂ (R 4)))) ∧
e.hom '' Set.range (fiberEmbedding t) =
PrimeSpectrum.zeroLocus (hyperplaneIdeal : Set (R 4))
/-- The defining-ideal orbit formulation captures exactly the geometric orbit
condition under all ambient automorphisms. Radicality follows from the specified
fiber algebra isomorphism, and is therefore not a new principal assumption. -/
theorem rectifiableFiber_iff_schemeRectifiable (t : ℂ)
(h : IsAffineThreeSpaceFiber t) :
IsRectifiableFiber t ↔ IsSchemeRectifiableFiber t := by
unfold IsRectifiableFiber IsSchemeRectifiableFiber fiberEmbedding
rw [AffineGeometry.quotientEmbedding_range]
exact AffineGeometry.ideal_transport_iff_scheme_transport (fiberIdeal t) hyperplaneIdeal
(fiberIdeal_isPrime_of_affineFiber t h).isRadical hyperplaneIdeal_isPrime.isRadical
/-- The fiber condition produces an isomorphism with affine three-space in the
category of schemes **over** `Spec ℂ`, not only an equivalence of sets. -/
theorem affineFiber_gives_schemeIso (t : ℂ) (h : IsAffineThreeSpaceFiber t) :
Nonempty (AffineGeometry.algebraSpec ℂ (FiberRing t) ≅
Over.mk (AffineSpace (Fin 3) (Spec (CommRingCat.of ℂ)) ↘
Spec (CommRingCat.of ℂ))) := by
obtain ⟨e⟩ := h
exact ⟨AffineGeometry.affineSpaceOverIsoOfAlgEquiv e⟩
/-- The full algebraic fiber package entails the full all-fiber geometric
conclusion, including closed immersion and nonrectifiability of its image. -/
theorem fiberPackage_gives_geometric_conclusion (h : FiberPackage) :
∀ t : ℂ,
Nonempty (AffineGeometry.algebraSpec ℂ (FiberRing t) ≅
Over.mk (AffineSpace (Fin 3) (Spec (CommRingCat.of ℂ)) ↘
Spec (CommRingCat.of ℂ))) ∧
IsClosedImmersion (fiberEmbedding t) ∧
¬ IsSchemeRectifiableFiber t := by
intro t
obtain ⟨haff, hnonrect⟩ := h t
refine ⟨affineFiber_gives_schemeIso t haff, inferInstance, ?_⟩
exact fun hrect => hnonrect ((rectifiableFiber_iff_schemeRectifiable t haff).mpr hrect)
end GeometricInterpretation
end StableCoordinate
end
end OAI
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