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Base-point cancellation for degree-zero divisors

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a compact connected Riemann surface, let $D = \sum_{p} n_{p} p$ be a divisor of degree zero and let $p_{0}, q_{0} \in X$ be two base points with point maps $u_{p_{0}}, u_{q_{0}}$ (The Abel-Jacobi map). Then in $Jac (X)$ $\sum_{p} n_{p} u_{p_{0}} (p) = \sum_{p} n_{p} u_{q_{0}} (p),$ so the class $u (D)$ is well defined without a base point; and for all $p, q \in X$ and all paths $γ$ from $p$ to $q$ , $u ((q) - (p)) = [ω \mapsto \int_{γ} ω], u ((q) - (p)) + u ((r) - (q)) = u ((r) - (p)) .$ In degree $1$ the base point does matter: for a single point $p$ one has $u_{p_{0}} (p) - u_{q_{0}} (p) = - u_{q_{0}} (p_{0}),$ which is nonzero in general: when $g \geq 1$ the map $u_{q_{0}}$ is an immersion, hence nonconstant, so $u_{q_{0}} (p_{0}) \neq 0$ for suitable $p_{0} \neq q_{0}$ .

Facts & Assumptions

Given: Full AC, a compact connected Riemann surface $X$ , two base points $p_{0}, q_{0}$ , and a degree-zero divisor $D = \sum_{p} n_{p} p$ .

[F1]

The addition rule $u_{b} (q) - u_{b} (p) = [ω \mapsto \int_{p}^{q} ω]$ holds for every base point $b$ and all $p, q \in X$ ; the point classes are represented by path integrals modulo the period lattice (The Abel-Jacobi map, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent, Path integral of a holomorphic differential on a Riemann surface).

[F2]

The linear extension $u_{b} (D) = \sum_{p} n_{p} u_{b} (p)$ is defined by finite sums, and on ${Div}^{0} (X)$ it is independent of the base point and additive (The Abel-Jacobi map, Divisors, principal divisors and canonical divisors on a Riemann surface).

[F3]

If $g \geq 1$ , then for every $p$ some holomorphic differential is nonzero at $p$ , so the derivative of $u_{q_{0}}$ at $p$ is nonzero and $u_{q_{0}}$ is an immersion; an immersion out of a connected surface is nonconstant, so there is $p_{0} \neq q_{0}$ with $u_{q_{0}} (p_{0}) \neq 0$ (Holomorphic differentials separate generic points, The Abel-Jacobi map is well defined and its degree-zero extension is base-point independent).

[F4]

Full AC is inherited from the Abel-Jacobi construction (The Axiom of Choice).

Verification

Given: The objects and conventions in the Statement.

1.1F1F2

By the addition rule of [F1] applied with base point $q_{0}$ , $u_{q_{0}} (p) = u_{q_{0}} (p_{0}) + [ω \mapsto \int_{p_{0}}^{p} ω] = u_{q_{0}} (p_{0}) + u_{p_{0}} (p)$ ; hence $u_{p_{0}} (p) - u_{q_{0}} (p) = - u_{q_{0}} (p_{0})$ for every $p$ , the displayed degree-one formula. Summing with coefficients $n_{p}$ gives $\sum_{p} n_{p} u_{p_{0}} (p) - \sum_{p} n_{p} u_{q_{0}} (p) = - (\sum_{p} n_{p}) u_{q_{0}} (p_{0}) = 0$ because $\deg D = \sum_{p} n_{p} = 0$ .

1.2F1

The formula $u ((q) - (p)) = [ω \mapsto \int_{γ} ω]$ is the addition rule of [F1], read for the difference of two points; it is independent of $γ$ by the well-definedness lemma [F1]. Adding the two classes for the pairs $(q, p)$ and $(r, q)$ and using additivity of the integral under concatenation gives $u ((q) - (p)) + u ((r) - (q)) = u ((r) - (p))$ .

2.1F1F3step 1.1

When $g \geq 1$ , [F3] makes $u_{q_{0}}$ nonconstant, while $u_{q_{0}} (q_{0}) = 0$ by [F1]. Hence there exists $p_{0} \neq q_{0}$ with $u_{q_{0}} (p_{0}) \neq 0$ . Step 1.1 then makes the degree-one difference $u_{p_{0}} (p) - u_{q_{0}} (p)$ nonzero for every $p$ . This proves the claimed base-point dependence in degree one without assuming a torus model.

3.1F4step 1.1step 1.2step 2.1∎

Claims: base-point independence on ${Div}^{0} (X)$ by step 1.1, the path-integral formula and three-point additivity by step 1.2, and the degree-one dependence by step 2.1; all under the inherited AC of [F4].

Depends on

Used by

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Sources

Eduard Looijenga, Riemann Surfaces (2007 author lecture notes) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Harvard Math 213b course notes (2026) (standard reference, not scraped)