Abel–Jacobi point separation on curves of positive genus #
On a proper regular integral curve of positive genus with exact constant field, linearly
equivalent closed points coincide whenever one has residue degree one. Thus the map
x ↦ [𝒪_X(x - x₀)] is injective on rational closed points, for any fixed closed point x₀.
The base point need not be rational for point separation; rationality is needed to make
these classes have degree zero.
This is the point-separation part of the Abel–Jacobi embedding theorem. The result concerns line-bundle classes, and does not construct a scheme morphism or prove an immersion. It applies over arbitrary exact constant fields, without perfectness assumptions.
At a rational closed point p, the space H⁰(X, 𝒪_X(p)) is one-dimensional in positive
genus, and the inclusion 𝒪_X ⟶ 𝒪_X(p) induces a bijection on global sections. These
results use Divisor.dim_ofPoint_eq_one_of_genus_pos through the scheme/function-field
comparison and supply the cohomological input to tangent separation.
References #
- R. Hartshorne, Algebraic Geometry, Chapter IV, Section 1 and Theorem 4.17.
- H. Stichtenoth, Algebraic Function Fields and Codes, 2nd ed., Theorem 1.6.8.
Linearly equivalent rational points on a curve of positive genus are equal. Here
X is a proper integral curve over k whose codimension-one local rings are discrete valuation
rings, with k integrally closed in k(X), and only the point x is required to have residue
degree one.
On a positive-genus proper regular curve with exact constants, the Abel–Jacobi line-bundle class separates rational closed points, with any fixed closed base point.
On a positive-genus proper regular curve with exact constants, allowing a simple pole at a rational closed point still gives only a one-dimensional space of global sections.
The inclusion 𝒪_X ⟶ 𝒪_X(p) is a bijection on zeroth cohomology when p is rational
and the curve has positive genus.