The Abel map from divisors to the relative Picard presheaf

Let f : X ⟶ S be a morphism of schemes. A relative effective Cartier divisor D on the base change X_T = T ×_S X has an invertible ideal sheaf 𝒪(-D), and its dual is the line bundle 𝒪(D). Sending D to the class of 𝒪(D) in Pic(X_T) / Pic(T) is natural in the scheme T over S: pulling back D along X_{T'} ⟶ X_T gives a relative effective Cartier divisor whose ideal sheaf is the pullback of 𝒪(-D) (Scheme.IdealSheafData.IsEffectiveCartier.pullback_mk_toInvertibleSheaf). This gives the Abel map D ↦ 𝒪(D) from the functor Div_{X/S} of relative effective Cartier divisors to the relative Picard presheaf T ↦ Pic(X_T) / Pic(T), as a natural transformation TauCeti.AlgebraicGeometry.abelMap. The empty divisor goes to the identity (TauCeti.AlgebraicGeometry.abelMap_app_top).

For a smooth proper curve X over a field, restricting to divisors of degree d gives the Abel maps Symᵈ X ⟶ Pic^d, and in degree one the degree-one Abel map X ⟶ Pic¹. A base point x₀ then gives the Abel–Jacobi map x ↦ 𝒪(x - x₀) into Pic⁰, by translating by 𝒪(-x₀). Separately, when f has a section, the target of the Abel map is also the rigidified Picard functor (TauCeti.AlgebraicGeometry.rigidifiedPicardFunctorIso). Neither the degree nor the representability of these functors is treated here.

The functor Div_{X/S} takes values in Type u, while line-bundle classes live in Type (u + 1), so the source of the Abel map is Div_{X/S} composed with uliftFunctor.

References

• S. Kleiman, The Picard scheme, in Fundamental Algebraic Geometry: Grothendieck's FGA Explained, Section 9.3 (the Abel map Div_{X/S} ⟶ Pic_{X/S}).
• The Stacks Project, Picard Schemes of Curves (Tag 0B95).

noncomputable def TauCeti.AlgebraicGeometry.abelMap {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) : (relativeEffectiveCartierSubfunctor f).toFunctor.comp CategoryTheory.uliftFunctor.{u + 1, u} ⟶ (relativePicardPresheaf f).comp (CategoryTheory.forget CommGrpCat)

The Abel map D ↦ 𝒪(D) of a morphism of schemes f : X ⟶ S: the natural transformation from the functor of relative effective Cartier divisors on the base changes X_T = T ×_S X to the relative Picard presheaf T ↦ Pic(X_T) / Pic(T), sending a divisor D to the class of the line bundle 𝒪(D), the dual of its ideal sheaf (abelMap_app_apply).

@[simp]

theorem TauCeti.AlgebraicGeometry.abelMap_app_apply {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) {T : (CategoryTheory.Over S)ᵒᵖ} (D : (relativeEffectiveCartierSubfunctor f).toFunctor.obj T) : (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := D } = ↑(LineBundleClass.mk ⋯.toInvertibleSheaf)⁻¹

The Abel map sends a relative effective Cartier divisor D on X_T to the image in Pic(X_T) / Pic(T) of the inverse of the class of its ideal sheaf, that is, of the class of 𝒪(D).

theorem TauCeti.AlgebraicGeometry.abelMap_app_top {S X : AlgebraicGeometry.Scheme} (f : X ⟶ S) (T : (CategoryTheory.Over S)ᵒᵖ) : (CategoryTheory.ConcreteCategory.hom ((abelMap f).app T)) { down := ⟨⊤, ⋯⟩ } = 1

The Abel map sends the empty divisor to the identity: 𝒪(∅) = 𝒪.