The Abel–Jacobi map separates field-valued points in positive genus

Let f : X ⟶ S be a smooth proper geometrically connected morphism of relative dimension one with a section x₀, and let ξ : Spec K ⟶ S be a point with values in a field K. If the fibre X_ξ = Spec K ×_S X has positive genus, then the Abel–Jacobi map x ↦ 𝒪(Γₓ - x₀) (TauCeti.AlgebraicGeometry.abelJacobiMap) is injective on the K-points of X over ξ. Over a field k this says that for a smooth proper geometrically connected curve X of genus g ≥ 1 with a rational point, the Abel–Jacobi map is injective on K-points for every field extension K / k, in particular on geometric points.

The proof works on the fibre X_ξ, a smooth proper integral curve over K with K algebraically closed in its function field. There the graph of a K-point x is the closed point p it hits, and its ideal sheaf is 𝒪(-p) (AlgebraicGeometry.Scheme.Hom.kerSheafIsoNegPoint). Since Pic(Spec K) is trivial, equal Abel–Jacobi classes make the ideal sheaves of the two graphs isomorphic, and rational points of a curve of positive genus are separated by their ideal sheaves (AlgebraicGeometry.Scheme.Hom.eq_of_mk_toInvertibleSheaf_eq_of_genus_pos).

This is the injectivity on geometric points in the statement that the Abel–Jacobi map of a curve of positive genus is a closed immersion. Neither the Jacobian scheme nor the injectivity of the Abel–Jacobi map on tangent vectors is treated here.

Main declarations

• TauCeti.AlgebraicGeometry.abelJacobiMap_app_injective_of_nontrivial_cohomology: the Abel–Jacobi map is injective on the points of X with values in a field over which the fibre has nonzero H¹(X_ξ, 𝒪);
• TauCeti.AlgebraicGeometry.abelJacobiMap_app_injective_of_genus_pos: for a curve of positive genus over a field k, the Abel–Jacobi map is injective on K-points for every field extension K / k.

References

• R. Hartshorne, Algebraic Geometry, Chapter IV, Section 1: on a curve of positive genus distinct points are not linearly equivalent.
• J. S. Milne, Jacobian varieties, in Arithmetic Geometry (G. Cornell, J. H. Silverman, eds.), Springer, 1986, Section 2: the canonical map Q ↦ 𝒪(Q - P) from a curve of positive genus to its Jacobian is a closed immersion.

theorem TauCeti.AlgebraicGeometry.abelJacobiMap_app_injective_of_nontrivial_cohomology {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) [AlgebraicGeometry.IsProper f] [AlgebraicGeometry.SmoothOfRelativeDimension 1 f] [AlgebraicGeometry.GeometricallyConnected f] {x₀ : S ⟶ X} (hx₀ : CategoryTheory.CategoryStruct.comp x₀ f = CategoryTheory.CategoryStruct.id S) {K : Type u} [Field K] (ξ : AlgebraicGeometry.Spec ↧K ⟶ S) (hg : Nontrivial (Scheme.Modules.Cohomology (InvertibleSheaf.trivial (CategoryTheory.Limits.pullback ξ f)).obj 1)) : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((abelJacobiMap f hx₀).app (Opposite.op (CategoryTheory.Over.mk ξ))))

The Abel–Jacobi map separates the points of a field-valued fibre of positive genus. Let f : X ⟶ S be a smooth proper geometrically connected morphism of relative dimension one with a section x₀, and let ξ : Spec K ⟶ S be a point with values in a field. If the first cohomology H¹(X_ξ, 𝒪) of the fibre X_ξ = Spec K ×_S X is nonzero, that is, if X_ξ has positive genus, then two K-points of X over ξ with the same class 𝒪(Γₓ - x₀) in Pic(X_ξ) / Pic(Spec K) are equal.

theorem TauCeti.AlgebraicGeometry.abelJacobiMap_app_injective_of_genus_pos {k : Type u} [Field k] {X : AlgebraicGeometry.Scheme} [X.Over (AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.IsProper (X ↘ AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.SmoothOfRelativeDimension 1 (X ↘ AlgebraicGeometry.Spec ↧k)] [AlgebraicGeometry.GeometricallyConnected (X ↘ AlgebraicGeometry.Spec ↧k)] {x₀ : AlgebraicGeometry.Spec ↧k ⟶ X} (hx₀ : CategoryTheory.CategoryStruct.comp x₀ (X ↘ AlgebraicGeometry.Spec ↧k) = CategoryTheory.CategoryStruct.id (AlgebraicGeometry.Spec ↧k)) (hg : 0 < AlgebraicGeometry.Scheme.genus k X) (K : Type u) [Field K] [Algebra k K] : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((abelJacobiMap (X ↘ AlgebraicGeometry.Spec ↧k) hx₀).app (Opposite.op (CategoryTheory.Over.mk (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap k K)))))))

The Abel–Jacobi map of a curve of positive genus separates points over every field extension. Let X be a smooth proper geometrically connected curve of genus g ≥ 1 over a field k, with a rational point x₀. For every field extension K / k, two K-points of X with the same class 𝒪(Γₓ - x₀) in Pic(X_K) / Pic(Spec K) are equal. Taking K algebraically closed, the Abel–Jacobi map is injective on geometric points.