The Abel–Jacobi maps of degree d #
Let f : X ⟶ S be a smooth proper morphism of relative dimension one with a section x₀. For
every natural number d, the functor Div^d_{X/S} of relative effective Cartier divisors of
degree d maps to the relative Picard presheaf T ↦ Pic(X_T) / Pic(T) by the Abel map
D ↦ 𝒪(D). Twisting by the d-th power of the class of the base-changed section x₀_T
normalizes it to the Abel–Jacobi map of degree d
abelJacobiMapOfDegree f hx₀ d : Div^d_{X/S} ⟶ (T ↦ Pic(X_T) / Pic(T)),
D ↦ 𝒪(D - d • x₀_T),
which sends the divisor d • x₀_T to the identity. Once Div^d_{X/S} is represented by the
symmetric power Symᵈ X (here this is known only for d ≤ 1), these become the maps
D ↦ 𝒪(D - d • x₀) out of Symᵈ X used in the classical construction of the Jacobian from
symmetric powers of a curve.
Addition of divisors is written multiplicatively in the monoid of relative effective Cartier
divisors (the sum D + E is cut out by the product of the ideal sheaves), so below D + E
and d • x₀_T are D * E and x₀_T ^ d in Lean.
This file proves the compatibilities of these maps with the operations on divisors:
- addition of divisors goes to the group law:
AJ_{d+e}(D + E) = AJ_d(D) · AJ_e(E)(abelJacobiMapOfDegree_app_mul); - adding copies of
x₀_Tdoes not change the class:AJ_{d+e}(D + e • x₀_T) = AJ_d(D)(abelJacobiMapOfDegree_app_mul_pow_basePoint), so the maps are compatible with the stabilization mapsDiv^d_{X/S} ⟶ Div^{d+e}_{X/S}; - in degree zero the map is trivial (
abelJacobiMapOfDegree_zero_app); - in degree one, under the isomorphism
Hom_S(-, X) ≅ Div¹_{X/S}given by graphs, it is the Abel–Jacobi mapx ↦ 𝒪(Γₓ - x₀_T)on points (whiskerRight_graphDivisorDegreeOne_comp_abelJacobiMapOfDegree_one); - the divisor
Γ_{x₁} + ⋯ + Γ_{x_d}ofdpoints goes to the product of the Abel–Jacobi classes of the points (abelJacobiMapOfDegree_app_prod_graphDivisor): onT-points, the composite of(x₁, …, x_d) ↦ Γ_{x₁} + ⋯ + Γ_{x_d}with the Abel–Jacobi map of degreedis the product ofdcopies of the Abel–Jacobi map, hence symmetric in the points.
Representability of Div^d_{X/S} for d ≥ 2, and the identification of the image with the
degree-zero part of the Picard functor, are not treated here.
References #
- S. Kleiman, The Picard scheme, in Fundamental Algebraic Geometry: Grothendieck's FGA
Explained, Section 9.3 (the Abel maps
Div^d_{X/S} ⟶ Pic_{X/S}). - J. S. Milne, Jacobian varieties, in Arithmetic Geometry (G. Cornell, J. H. Silverman,
eds.), Springer, 1986, Sections 4 and 5 (the maps
D ↦ 𝒪(D - d P)fromSymᵈ Xto the Jacobian).
The Abel–Jacobi map of degree d of a smooth proper morphism f : X ⟶ S of relative
dimension one with a section x₀: the natural transformation from the functor Div^d_{X/S} of
relative effective Cartier divisors of degree d 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
𝒪(D - d • x₀_T) = 𝒪(D) ⊗ 𝒪(x₀_T)^{⊗ -d}, where x₀_T is the base change of x₀
(abelJacobiMapOfDegree_app_apply).
Instances For
The Abel–Jacobi map of degree d sends a divisor D to the quotient of the Abel map at D
by the d-th power of the Abel map at the graph of the base point, that is, to the class of
𝒪(D - d • x₀_T).
The Abel–Jacobi maps are additive: the sum of divisors D and E of degrees d and
e goes to the product of their classes, 𝒪(D + E - (d + e) • x₀_T) = 𝒪(D - d • x₀_T) ⊗ 𝒪(E - e • x₀_T).
In degree zero the Abel–Jacobi map is trivial: the only divisor of degree zero is the empty
divisor, and 𝒪(∅) = 𝒪.
The divisor n • x₀_T, the n-th multiple of the graph of the base point, has degree n.
The Abel–Jacobi map of degree d sends d • x₀_T to the identity:
𝒪(d • x₀_T - d • x₀_T) = 𝒪.
The Abel–Jacobi maps are compatible with stabilization: adding e copies of the base
point to a divisor D of degree d does not change its class,
𝒪(D + e • x₀_T - (d + e) • x₀_T) = 𝒪(D - d • x₀_T).
In degree one the Abel–Jacobi map is the Abel–Jacobi map on points: composing the
isomorphism Hom_S(-, X) ≅ Div¹_{X/S}, x ↦ Γₓ, with the Abel–Jacobi map of degree one gives
the Abel–Jacobi map x ↦ 𝒪(Γₓ - x₀_T).
The sum Γ_{x_i}, i ∈ s of the graphs of a finite family of T-points has degree the number
of points.
The Abel–Jacobi map of a sum of points is the product of their Abel–Jacobi classes: for
a finite family of T-points x_i, i ∈ s, the divisor ∑ i ∈ s, Γ_{x_i} goes to
∏ i ∈ s, 𝒪(Γ_{x_i} - x₀_T).