| import OAI.NumberTheory.PiExponent.Geometry.CurveGlobalMonomial |
| import OAI.NumberTheory.PiExponent.Geometry.CurvePlaceCenter |
| import OAI.NumberTheory.PiExponent.Geometry.ProjectiveLatticePullback |
| import OAI.NumberTheory.PiExponent.Geometry.ProjectiveSpaceBasics |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace PiExponent.ProjectiveLocalCoefficients |
| open AlgebraicGeometry CategoryTheory MvPolynomial |
| open PiExponentSeshadri.Projective |
| attribute [local instance] MvPolynomial.gradedAlgebra |
| variable {R A B σ : Type} [CommRing R] [CommRing A] [CommRing B] |
| |
| theorem normalizedMap_natural (h : A →+* B) (k : R →+* A) |
| (y : σ → A) (i : σ) (hi : y i = 1) : |
| Spec.map (CommRingCat.ofHom h) ≫ normalizedMap k y i hi = |
| normalizedMap (h.comp k) (fun j => h (y j)) i (by rw [hi, map_one]) := by |
| unfold normalizedMap |
| rw [fromUnitCoordinate_natural] |
| congr 1 |
| exact coordinates_eval_natural h k y |
|
|
| theorem normalizedMap_scale (k : R →+* A) (a b : σ → A) |
| (i j : σ) (hi : a i = 1) (hj : b j = 1) |
| (c : Aˣ) (hscale : ∀ q, b q = (c : A) * a q) : |
| normalizedMap k a i hi = normalizedMap k b j hj := by |
| let f := eval₂Hom k a |
| let g := eval₂Hom k b |
| have hf : IsUnit (f (MvPolynomial.X (R := R) i)) := by |
| simpa only [f, eval₂Hom_X', hi] using (isUnit_one : IsUnit (1 : A)) |
| have hg : IsUnit (g (MvPolynomial.X (R := R) i)) := by |
| simpa only [g, eval₂Hom_X', hscale i, hi, mul_one] using c.isUnit |
| have hg' : IsUnit (g (MvPolynomial.X (R := R) j)) := by |
| simpa only [g, eval₂Hom_X', hj] using (isUnit_one : IsUnit (1 : A)) |
| have H : ∀ n p, p ∈ homogeneousSubmodule σ R n → |
| g p = (c : A) ^ n * f p := by |
| intro n p hp |
| change eval₂ k b p = _ |
| have hb : b = fun q => (c : A) * a q := funext hscale |
| rw [hb] |
| exact homogeneous_eval₂_scale hp k a c |
| unfold normalizedMap |
| exact (fromUnitCoordinate_scale (𝒜 := homogeneousSubmodule σ R) (d := 1) f (by decide) (coordinates_X_mem (K := R) i) |
| hf g hg c H).trans |
| (fromUnitCoordinate_eq (𝒜 := homogeneousSubmodule σ R) (d := 1) (e := 1) g (by decide) (by decide) |
| (coordinates_X_mem (K := R) i) (coordinates_X_mem (K := R) j) hg hg') |
| |
| end PiExponent.ProjectiveLocalCoefficients |
| |
| namespace PiExponent.CurveGlobalMonomialLocal |
| open AlgebraicGeometry CategoryTheory MvPolynomial |
| open ProjectiveLatticePullback ProjectiveLocalCoefficients WeightedSectionOrder |
| open WeightedLocalLattice PiExponentSeshadri.Projective |
| open scoped BigOperators |
| attribute [local instance] MvPolynomial.gradedAlgebra |
| |
| theorem latticeProjectiveMap_generic |
| {F A E σ : Type} [CommRing F] [CommRing A] [IsDomain A] |
| [Field E] [Algebra A E] [IsFractionRing A E] [Fintype σ] |
| (k : F →+* A) (y : σ → E) (i : σ) (hi : y i ≠ 0) |
| (hspan : generatedLattice A y Finset.univ = Submodule.span A {y i}) |
| (z : σ) (hz : y z = 1) : |
| Spec.map (CommRingCat.ofHom (algebraMap A E)) ≫ latticeProjectiveMap k y i hi hspan = |
| normalizedMap ((algebraMap A E).comp k) y z hz := by |
| rw [latticeProjectiveMap, normalizedMap_natural] |
| apply normalizedMap_scale _ _ y i z _ hz (Units.mk0 (y i) hi) |
| intro b |
| change y b = y i * algebraMap A E (latticeCoordinates y i hi hspan b) |
| rw [show algebraMap A E (latticeCoordinates y i hi hspan b) = y b / y i from |
| coordinate_coefficient_image y Finset.univ i (Finset.mem_univ i) hi hspan b (Finset.mem_univ b)] |
| field_simp |
| |
| def monomialCoordinates {F E ι σ : Type} [CommRing F] [CommRing E] [Algebra F E] |
| (a : σ → ι →₀ ℕ) (x : ι → E) (s : σ) : E := |
| eval₂Hom (algebraMap F E) x (monomial (a s) 1) |
| |
| theorem monomialCoordinates_eq_prod |
| {F E ι σ : Type} [CommRing F] [CommRing E] [Algebra F E] [Fintype ι] |
| (a : σ → ι →₀ ℕ) (x : ι → E) (s : σ) : |
| monomialCoordinates (F := F) a x s = ∏ j, x j ^ (a s j) := by |
| rw [monomialCoordinates, eval₂Hom_monomial, map_one, one_mul] |
| exact Finsupp.prod_fintype _ _ (fun _ => pow_zero _) |
| |
| @[simp] theorem monomialCoordinates_zero |
| {F E ι σ : Type} [CommRing F] [CommRing E] [Algebra F E] |
| (a : σ → ι →₀ ℕ) (x : ι → E) (z : σ) (hz : a z = 0) : |
| monomialCoordinates (F := F) a x z = 1 := by |
| simp [monomialCoordinates, hz] |
| |
| theorem genericMonomialMap_comp_projectiveMonomialMap |
| {F E ι σ : Type} [CommRing F] [CommRing E] [Algebra F E] |
| (a : σ → ι →₀ ℕ) (z : σ) (hz : a z = 0) |
| (coordinate : ι → σ) (hcoordinate : ∀ i, a (coordinate i) = Finsupp.single i 1) |
| (x : ι → E) : |
| CurveMonomialMap.genericMonomialMap a z hz coordinate hcoordinate x ≫ |
| WeightedCompactification.projectiveMonomialMap a = |
| normalizedMap (algebraMap F E) (monomialCoordinates (F := F) a x) z |
| (monomialCoordinates_zero a x z hz) := by |
| have heval : (eval₂Hom (algebraMap F E) x).comp |
| (WeightedCompactification.affineMonomialMap (R := F) a) = |
| eval₂Hom (algebraMap F E) (monomialCoordinates (F := F) a x) := by |
| apply MvPolynomial.ringHom_ext |
| · intro r |
| simp [WeightedCompactification.affineMonomialMap] |
| · intro b |
| simp [WeightedCompactification.affineMonomialMap, monomialCoordinates] |
| rw [CurveMonomialMap.genericMonomialMap, Category.assoc, |
| WeightedCompactification.affineChartMap_comp_projectiveMonomialMap, |
| fromUnitCoordinate_natural] |
| unfold normalizedMap |
| congr 1 |
| |
| theorem centerMorphism_eq_latticeProjectiveMap |
| {F E σ : Type} [Field F] [Field E] [Algebra F E] [Fintype σ] |
| (f : E) (hf : Transcendental F f) |
| [FiniteDimensional (IntermediateField.adjoin F {f}) E] |
| (g : CurveNormalizationModel.parameterCurve f hf ⟶ ProjectiveO1.projectiveSpace F σ) |
| (y : σ → E) (z : σ) (hz : y z = 1) |
| (hg : CurveNormalizationModel.parameterCurveGenericPoint f hf ≫ g = |
| normalizedMap (algebraMap F E) y z hz) |
| (p : CurveValuationCenter.NormalizedPlace F E) (i : σ) (hi : y i ≠ 0) |
| (hspan : generatedLattice (PlaceValuationRing.ring p) y Finset.univ = |
| Submodule.span (PlaceValuationRing.ring p) {y i}) : |
| CurvePlaceCenter.centerMorphism f hf p ≫ g = |
| latticeProjectiveMap (algebraMap F (PlaceValuationRing.ring p)) y i hi hspan := by |
| apply CurvePlaceCenter.centerMorphism_comp_eq_of_generic f hf p |
| rw [latticeProjectiveMap_generic _ y i hi hspan z hz, hg] |
| rfl |
| |
| theorem centerMorphism_globalMonomialMap |
| {F E ι σ : Type} [Field F] [Field E] [Algebra F E] [Fintype σ] |
| (f : E) (hf : Transcendental F f) |
| [FiniteDimensional (IntermediateField.adjoin F {f}) E] |
| (a : σ → ι →₀ ℕ) (z : σ) (hz : a z = 0) |
| (coordinate : ι → σ) (hcoordinate : ∀ j, a (coordinate j) = Finsupp.single j 1) |
| (x : ι → E) |
| (g : CurveNormalizationModel.parameterCurve f hf ⟶ |
| Proj (WeightedCompactification.imageGrade (R := F) a)) |
| (hg : CurveNormalizationModel.parameterCurveGenericPoint f hf ≫ g = |
| CurveMonomialMap.genericMonomialMap a z hz coordinate hcoordinate x) |
| (p : CurveValuationCenter.NormalizedPlace F E) (i : σ) |
| (hi : monomialCoordinates (F := F) a x i ≠ 0) |
| (hspan : generatedLattice (PlaceValuationRing.ring p) |
| (monomialCoordinates (F := F) a x) Finset.univ = |
| Submodule.span (PlaceValuationRing.ring p) {monomialCoordinates (F := F) a x i}) : |
| CurvePlaceCenter.centerMorphism f hf p ≫ |
| (g ≫ WeightedCompactification.projectiveMonomialMap a) = |
| latticeProjectiveMap (algebraMap F (PlaceValuationRing.ring p)) |
| (monomialCoordinates (F := F) a x) i hi hspan := by |
| apply centerMorphism_eq_latticeProjectiveMap f hf _ _ z |
| (monomialCoordinates_zero a x z hz) _ p i hi hspan |
| rw [← Category.assoc, hg] |
| exact genericMonomialMap_comp_projectiveMonomialMap a z hz coordinate hcoordinate x |
| |
| end PiExponent.CurveGlobalMonomialLocal |
| |
| end |
| |
| end OAI |
| |
