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The Picard group of divisor classes and its degree-zero part

Statement

Assume the full Axiom of Choice (The Axiom of Choice). Let $X$ be a compact connected Riemann surface and let $Div (X)$ be its group of divisors (Divisors, principal divisors and canonical divisors on a Riemann surface). Let $Prin (X) : = {(f) : f \in M (X)^{*}}$ be the subgroup of principal divisors. Divisors are linearly equivalent when $D \sim D^{'}$ iff $D - D^{'} \in Prin (X)$ . The quotient $Pic (X) : = Div (X) / Prin (X)$ is the Picard group of divisor classes. Since principal divisors have degree zero, degree descends to a homomorphism $\deg : Pic (X) \to Z$ , and ${Pic}^{0} (X) : = \ker (\deg) = {Div}^{0} (X) / Prin (X), {Div}^{0} (X) : = \ker (\deg : Div (X) \to Z),$ is the subgroup of degree-zero divisor classes. The map $[D] \mapsto [O (D)]$ is a canonical group isomorphism from $Pic (X)$ to the isomorphism classes of holomorphic line bundles on $X$ under tensor product (The holomorphic line bundle associated to a divisor, Holomorphic line bundles and meromorphic sections on a Riemann surface, Every holomorphic line bundle on a compact Riemann surface has a meromorphic section). In particular, $O (D) ≅ O (D^{'})$ iff $D \sim D^{'}$ , and ${Pic}^{0} (X)$ corresponds exactly to degree-zero line bundles, with $\deg (O (D)) : = \deg D$ . Full AC is used through the meromorphic-section existence lemma for surjectivity; the construction of $O (D)$ on compact $X$ uses a finite cover.

Facts & Assumptions

Given: Full AC and a compact connected Riemann surface $X$ .

[F1]

For nonzero meromorphic functions, $(f g) = (f) + (g)$ and $(1 / f) = - (f)$ , so principal divisors form a subgroup of $Div (X)$ (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F2]

On compact $X$ , every principal divisor has degree zero, and divisor degree is additive (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

The divisor-bundle construction satisfies $O (D + D^{'}) ≅ O (D) \otimes O (D^{'})$ and $O (0) ≅ X \times C$ (The holomorphic line bundle associated to a divisor).

[F4]

If $D \sim D^{'}$ , then $O (D) ≅ O (D^{'})$ ; the supplier constructs this isomorphism from a meromorphic function whose divisor is $D^{'} - D$ (The holomorphic line bundle associated to a divisor).

[F5]

The bundle $O (D)$ has a canonical meromorphic section $s_{D}$ with divisor $(s_{D}) = D$ , obtained locally from equations $f_{i}$ of $D$ and frames $e_{i}$ by $s_{D} ∣_{U_{i}} = f_{i} e_{i}$ (The holomorphic line bundle associated to a divisor).

[F6]

A meromorphic section is a family of meromorphic local coefficients satisfying the same frame transition law (Holomorphic line bundles and meromorphic sections on a Riemann surface).

[F7]

Every holomorphic line bundle on compact connected $X$ has a nonzero meromorphic section; for each prescribed $p \in X$ one can choose it holomorphic off $p$ with a pole at $p$ (Every holomorphic line bundle on a compact Riemann surface has a meromorphic section).

[F8]

Full AC is assumed and supplies the hypothesis used by [F7]; no choice is used in forming the divisor quotient or its degree kernel (The Axiom of Choice).

[F9]

A nonzero meromorphic function has a local factorization $f = z^{m} u$ with $u$ holomorphic and nonzero; order zero therefore means a holomorphic unit (Divisors, principal divisors and canonical divisors on a Riemann surface).

[F10]

The divisor of a nonzero meromorphic section is locally finite and has finite support on compact $X$ (Holomorphic line bundles and meromorphic sections on a Riemann surface).

Proof

technique · divisor quotient and local line-bundle isomorphisms

1.1F1givenalgebra

Let $Prin (X) = {(f) : f \in M (X)^{*}}$ . By [F1] it is a subgroup of the abelian group $Div (X)$ . Thus $D \sim D^{'}$ exactly when $D - D^{'} \in Prin (X)$ is an equivalence relation, and its classes form the quotient abelian group $Pic (X) = Div (X) / Prin (X)$ .

2.1F2step 1.1algebra

Divisor degree is additive and vanishes on $Prin (X)$ by [F2], so it induces a homomorphism $\deg : Pic (X) \to Z$ . A class $[D]$ lies in its kernel exactly when $\deg D = 0$ , hence $\ker (\deg) = {Div}^{0} (X) / Prin (X)$ , which is the stated ${Pic}^{0} (X)$ .

2.2F3F4step 1.1algebra

Define $Θ : Pic (X) \to {Pic}_{b u n} (X)$ by $Θ ([D]) = [O (D)]$ , where ${Pic}_{b u n} (X)$ is the group of isomorphism classes of holomorphic line bundles under tensor product. This is well defined by [F4], and [F3] makes it a group homomorphism.

3.1F5F6step 2.2givenalgebra

Suppose $O (D) ≅ O (D^{'})$ , and let $Φ$ be a holomorphic bundle isomorphism between them. By [F5], $Φ (s_{D})$ and $s_{D^{'}}$ are nonzero meromorphic sections; a bundle isomorphism is locally multiplication by a nowhere-zero holomorphic function, so $Φ (s_{D})$ has divisor $D$ . By [F6], their ratio $f : = Φ (s_{D}) / s_{D^{'}}$ is a global nonzero meromorphic function, with $(f) = D - D^{'}$ . Thus $D \sim D^{'}$ and $Θ$ is injective.

4.1F5F6F7F8F9F10step 2.2step 3.1construct∎

Let $E \to X$ be any holomorphic line bundle. By [F7] choose a nonzero meromorphic section $s$ and put $D = (s)$ , a finite divisor by [F10]. On a common finite refinement of the local frames for $E$ and the equations defining $O (D)$ , write $s = h_{i} v_{i}$ and $s_{D} = f_{i} e_{i}$ . Since $(h_{i}) = (f_{i}) = D ∣_{U_{i}}$ , [F9] makes $a_{i} : = f_{i} / h_{i}$ holomorphic and nowhere zero. Define $Φ (v_{i}) = a_{i} e_{i}$ ; the transition laws $g_{i j}^{E} = h_{i} / h_{j}$ and $g_{i j}^{O} = f_{i} / f_{j}$ give $a_{j} g_{i j}^{O} = g_{i j}^{E} a_{i}$ , so these local maps glue to a holomorphic line-bundle isomorphism $E \to O (D)$ carrying $s$ to $s_{D}$ . Hence $Θ$ is surjective. Together with step 3.1 this proves the claimed group isomorphism; degree is well defined on line-bundle classes by injectivity, and ${Pic}^{0} (X)$ corresponds exactly to degree-zero line bundles. Full AC is used here only through [F7]; the compact divisor-bundle construction uses a finite cover.

Depends on

Used by

Picard zero is the Jacobian Corollary

Dependency tree · two levels

43 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Karl Otto Forster, Lectures on Riemann Surfaces (GTM 81, Springer 1981), translated by Bruce Gilligan (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)