The Abel–Jacobi maps land in Pic⁰ #
Let f : X ⟶ S be a smooth proper morphism of relative dimension one with geometrically
connected fibres. A relative effective Cartier divisor D on X_T = T ×_S X restricts, over a
point ξ : Spec Ω ⟶ T with values in a field, to an effective Cartier divisor D_ξ on the fibre,
which is a connected smooth proper curve over Ω; the line bundle 𝒪(D) restricts to 𝒪(D_ξ),
whose degree is the rank of D_ξ over Ω
(Scheme.IdealSheafData.IsEffectiveCartier.eulerDegree_inv_mk_toInvertibleSheaf_of_smooth).
Since the rank of D is preserved by base change, the fibre degree of the Abel map D ↦ 𝒪(D) at
ξ is the rank of D at the image of ξ (relativePicardFiberDegree_abelMap_app), and it is d
for a divisor in Div^d_{X/S}.
Consequently, for a section x₀, the Abel–Jacobi map of degree d, D ↦ 𝒪(D - d • x₀_T), takes
values in the degree-zero Picard functor Pic⁰_{X/S}: both D and d • x₀_T have degree d on
every fibre. This gives the normalized Abel maps Div^d_{X/S} ⟶ Pic⁰_{X/S}
(abelJacobiMapOfDegreeToPicZero) and, in degree one, the Abel–Jacobi map x ↦ 𝒪(Γₓ - x₀_T)
from the functor of points of X to Pic⁰_{X/S} (abelJacobiMapToPicZero), on functors of
points. Representability of Pic⁰_{X/S} is not treated here.
Main declarations #
TauCeti.AlgebraicGeometry.relativePicardFiberDegree_abelMap_app: the fibre degree of𝒪(D)is the rank ofD;TauCeti.AlgebraicGeometry.abelJacobiMapOfDegree_app_mem_relativePicardDegreeZeroSubfunctor_obj:𝒪(D - d • x₀_T)has degree zero on every fibre;TauCeti.AlgebraicGeometry.abelJacobiMapOfDegreeToPicZero: the Abel–Jacobi map of degreedas a natural transformationDiv^d_{X/S} ⟶ Pic⁰_{X/S}into the degree-zero subfunctor;TauCeti.AlgebraicGeometry.abelJacobiMapToPicZero: the Abel–Jacobi mapHom_S(-, X) ⟶ Pic⁰_{X/S}.
References #
- S. Kleiman, The Picard scheme, in Fundamental Algebraic Geometry: Grothendieck's FGA
Explained, Section 9.3 (the Abel maps) and Section 9.5 (
Pic⁰). - J. S. Milne, Jacobian varieties, in Arithmetic Geometry (G. Cornell, J. H. Silverman,
eds.), Springer, 1986, Sections 2 and 5 (the maps
D ↦ 𝒪(D - d P)into the Jacobian).
The fibre degree of the Abel map is the degree of the divisor. For a relative effective
Cartier divisor D on X_T and a point ξ : Spec Ω ⟶ T with values in a field, the line bundle
𝒪(D) has degree on the fibre over ξ equal to the rank of D over T at the image of ξ.
A relative effective Cartier divisor of degree d has fibre degree d under the Abel map:
𝒪(D) has degree d on every fibre.
The Abel–Jacobi map of degree d takes values in Pic⁰: for a divisor D of degree d,
the class of 𝒪(D - d • x₀_T) has degree zero on every geometric fibre.
The image of the Abel–Jacobi map of degree d lies in the degree-zero subfunctor
Pic⁰_{X/S} of the relative Picard presheaf.
The Abel–Jacobi map of degree d into Pic⁰_{X/S}: the natural transformation from the
functor Div^d_{X/S} of relative effective Cartier divisors of degree d to the subfunctor
Pic⁰_{X/S} of classes of degree zero on every geometric fibre, sending D to the class of
𝒪(D - d • x₀_T). Followed by the inclusion of Pic⁰_{X/S} it is abelJacobiMapOfDegree
(abelJacobiMapOfDegreeToPicZero_ι).
Instances For
The Abel–Jacobi map of degree d into Pic⁰_{X/S}, followed by the inclusion of
Pic⁰_{X/S} into the relative Picard presheaf, is the Abel–Jacobi map of degree d.
The Abel–Jacobi map of degree d into Pic⁰_{X/S}, followed by the inclusion of
Pic⁰_{X/S} into the relative Picard presheaf, is the Abel–Jacobi map of degree d.
The class underlying the degree-d Abel–Jacobi map into Pic⁰ is its value in the relative
Picard presheaf.
The Abel–Jacobi map into Pic⁰_{X/S}: the natural transformation from the functor of
points T ↦ Hom_S(T, X) to the subfunctor Pic⁰_{X/S} of classes of degree zero on every
geometric fibre, sending a T-point x to the class of 𝒪(Γₓ - x₀_T). Followed by the inclusion
of Pic⁰_{X/S} it is abelJacobiMap (abelJacobiMapToPicZero_ι).
Instances For
The Abel–Jacobi map into Pic⁰ is the degree-one normalized Abel map applied to the graph
divisor of a point.
The Abel–Jacobi map into Pic⁰_{X/S}, followed by the inclusion of Pic⁰_{X/S} into the
relative Picard presheaf, is the Abel–Jacobi map x ↦ 𝒪(Γₓ - x₀_T).
The Abel–Jacobi map into Pic⁰_{X/S}, followed by the inclusion of Pic⁰_{X/S} into the
relative Picard presheaf, is the Abel–Jacobi map x ↦ 𝒪(Γₓ - x₀_T).
The class underlying the Abel–Jacobi map into Pic⁰ is its value in the relative Picard
presheaf.
The class underlying the Abel–Jacobi map into Pic⁰_{X/S} sends the base point to the
identity.