| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.ClosedStalk |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.IntegralCurveIdeal |
| import OAI.AlgebraicGeometry.Seshadri.Sheaves.AffineOrderRestriction |
|
|
| namespace OAI |
|
|
|
|
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open MaximalSeshadri.ProjectiveBertini |
|
|
| lemma IntegralCurve.point_in_range_of_affine_ideal (S : Surface) (C : IntegralCurve S) |
| (U : S.scheme.affineOpens) |
| (ρ : letI := (openScalars S.structureMap U.1).toAlgebra; |
| Γ(S.scheme,U.1) →ₐ[ℂ] ℂ) |
| (h : C.embedding.ker.ideal U ≤ RingHom.ker ρ) : |
| (affineComplexPoint S.structureMap U ρ).left (fieldPoint ℂ) ∈ Set.range C.embedding := by |
| let q : PrimeSpectrum Γ(S.scheme,U.1) := ⟨RingHom.ker ρ,RingHom.ker_isPrime _⟩ |
| have hh : U.2.fromSpec q ∈ (C.embedding.ker.support : Set S.scheme) ∩ U.1 := by |
| rw [C.embedding.ker.coe_support_inter U,← U.2.fromSpec_image_zeroLocus] |
| exact ⟨q,h,rfl⟩ |
| have he : (C.embedding.ker.support : Set S.scheme) = Set.range C.embedding := by |
| rw [Scheme.Hom.support_ker] |
| exact C.embedding.isClosedEmbedding.isClosed_range.closure_eq |
| exact he ▸ hh.1 |
|
|
| theorem IntegralCurve.affine_ideal_not_le_next_multiplicity (S : Surface) |
| (C : IntegralCurve S) (U : S.scheme.affineOpens) |
| (ρ : letI := (openScalars S.structureMap U.1).toAlgebra; |
| Γ(S.scheme,U.1) →ₐ[ℂ] ℂ) : |
| ¬ C.embedding.ker.ideal U ≤ (RingHom.ker ρ)^ |
| (curveMultiplicity S C (affineComplexPoint S.structureMap U ρ)+1) := by |
| classical |
| let := (openScalars S.structureMap U.1).toAlgebra |
| let p : ComplexPoint S := affineComplexPoint S.structureMap U ρ |
| change ¬ C.embedding.ker.ideal U ≤ (RingHom.ker ρ)^(curveMultiplicity S C p+1) |
| unfold curveMultiplicity |
| split_ifs with h |
| · let q := h.choose |
| have hq : C.embedding q = p.image := h.choose_spec |
| have hpU : p.image ∈ U.1 := by |
| have H := Set.mem_range_self (f := U.2.fromSpec) |
| (⟨RingHom.ker ρ,RingHom.ker_isPrime _⟩ : PrimeSpectrum Γ(S.scheme,U.1)) |
| rw [U.2.range_fromSpec] at H |
| exact H |
| have hqu : C.embedding q ∈ U.1 := hq ▸ hpU |
| let : Nonempty U.1 := ⟨⟨_,hpU⟩⟩ |
| let : Nonempty (C.embedding ⁻¹ᵁ U.1) := ⟨⟨q,hqu⟩⟩ |
| let : IsNoetherianRing Γ(S.scheme,U.1) := IsLocallyNoetherian.component_noetherian U |
| let I := C.embedding.ker.ideal U |
| have hI : I ≠ ⊥ := C.affine_ideal_ne_bot S U inferInstance |
| have hf : {n : ℕ | ¬ I ≤ (RingHom.ker ρ)^(n+1)}.Nonempty := by |
| by_contra hn |
| have hn' (n : ℕ) : I ≤ (RingHom.ker ρ)^(n+1) := by |
| by_contra hh |
| exact hn ⟨n,hh⟩ |
| apply hI |
| apply bot_unique |
| intro a ha |
| have hall : a ∈ (⨅ n : ℕ, (RingHom.ker ρ)^n) := by |
| apply Ideal.mem_iInf.mpr |
| intro n |
| cases n with |
| | zero => simp |
| | succ n => exact hn' n ha |
| rw [(RingHom.ker ρ).iInf_pow_eq_bot_of_isDomain (RingHom.ker_ne_top _)] at hall |
| exact hall |
| have he (n : ℕ) : RingHom.ker (C.embedding.stalkMap q).hom ≤ |
| IsLocalRing.maximalIdeal (S.scheme.presheaf.stalk (C.embedding q))^(n+1) ↔ |
| I ≤ (RingHom.ker ρ)^(n+1) := by |
| rw [closedImmersion_stalk_kernel C.embedding U q hqu,Ideal.map_le_iff_le_comap] |
| rw [affine_point_germ_power S.structureMap U ρ (C.embedding q) hqu hq.symm (n+1)] |
| have horder : idealOrder (RingHom.ker (C.embedding.stalkMap q).hom) = |
| sInf {n : ℕ | ¬ I ≤ (RingHom.ker ρ)^(n+1)} := by |
| unfold idealOrder |
| congr 1 |
| ext n |
| exact not_congr (he n) |
| rw [horder] |
| exact Nat.sInf_mem hf |
| · simp only [Nat.zero_add,pow_one] |
| intro hi |
| exact h (C.point_in_range_of_affine_ideal S U ρ hi) |
|
|
| end |
| end MaximalSeshadri.Geometry |
|
|
| end |
|
|
|
|
| end OAI |
|
|
