@[reducible, inline]

noncomputable abbrev Lec17JacobianNormalization.SpecC : AlgebraicGeometry.Scheme

The affine scheme Spec ℂ, used as the base for varieties over the complex numbers.

structure Lec17JacobianNormalization.SmoothProjectiveCurve : Type 1

A smooth projective curve over ℂ: a proper, smooth, integral scheme over Spec ℂ of relative dimension 1, together with its genus.

• toScheme : AlgebraicGeometry.Scheme
• structureMorphism : self.toScheme ⟶ SpecC
• isProper : AlgebraicGeometry.IsProper self.structureMorphism
• isSmoothOfRelDim1 : AlgebraicGeometry.SmoothOfRelativeDimension 1 self.structureMorphism
• isIntegral : AlgebraicGeometry.IsIntegral self.toScheme
• genus : ℕ

structure Lec17JacobianNormalization.AbelianGroupScheme : Type 1

An abelian group scheme over ℂ: a proper smooth scheme over Spec ℂ equipped with an identity section. (Group operations are not packaged here; this captures the geometric side used in the Abel--Jacobi statement.)

• toScheme : AlgebraicGeometry.Scheme
• structureMorphism : self.toScheme ⟶ SpecC
• isProper : AlgebraicGeometry.IsProper self.structureMorphism
• isSmooth : AlgebraicGeometry.Smooth self.structureMorphism
• identitySection : SpecC ⟶ self.toScheme
• identitySection_comp : CategoryTheory.CategoryStruct.comp self.identitySection self.structureMorphism = CategoryTheory.CategoryStruct.id SpecC

theorem Lec17JacobianNormalization.proposition_26_jacobian_variety (X : SmoothProjectiveCurve) : ∃ (JacX : AbelianGroupScheme) (AJmap : X.toScheme ⟶ JacX.toScheme), CategoryTheory.CategoryStruct.comp AJmap JacX.structureMorphism = X.structureMorphism ∧ AlgebraicGeometry.SmoothOfRelativeDimension X.genus JacX.structureMorphism

Proposition 26 (Lecture 17, Abel--Jacobi). Any smooth projective curve X of genus g admits a Jacobian variety Jac(X), an abelian group scheme of relative dimension g, together with an Abel--Jacobi morphism X → Jac(X) over Spec ℂ.

noncomputable def Lec17JacobianNormalization.lemma_28_birational_surj_normal_iso (B : Type u_1) (A : Type u_2) (K : Type u_3) [CommRing B] [IsDomain B] [CommRing A] [Field K] [Algebra B K] [IsFractionRing B K] [Algebra A K] [Algebra B A] [IsScalarTower B A K] [IsIntegrallyClosed B] [IsIntegralClosure A B K] : A ≃ₐ[B] B

Lemma 28 (Lecture 17). If B is an integrally closed domain with fraction field K and A is the integral closure of B in K, then A ≃ₐ[B] B: an integrally closed domain is already its own normalization.

theorem Lec17JacobianNormalization.locallyQuasiFinite_of_isProper_isDominant_isIso_fromNormalization {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [AlgebraicGeometry.IsProper f] [AlgebraicGeometry.IsDominant f] [CategoryTheory.IsIso (AlgebraicGeometry.Scheme.Hom.fromNormalization f)] : AlgebraicGeometry.LocallyQuasiFinite f

A proper dominant morphism that becomes an isomorphism after passing to the normalization of the source is locally quasi-finite. This is the key geometric input used to upgrade normalization isomorphisms to finite morphisms in Lecture 17.