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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Degree-zero Picard group as points of an abelian variety

PDF p. 13: Picard group of a smooth affine curve

Statement

For a smooth projective integral curve X over an algebraically closed field k, the group of degree-zero line-bundle classes Pic⁰(X) is isomorphic to J(k), where J is an abelian variety over k.

A line bundle is a locally free module of rank one; Pic⁰(X) is the kernel of the degree homomorphism on its tensor-product Picard group. An abelian variety is a proper geometrically integral group scheme of finite type over k; J(k) is its group of k-rational points.

External input

This is the stated degree-zero Picard variety identification in the cited opening paragraphs, restricted to integral curves. It is imported from the literature; no local reconstruction of its construction or proof is claimed.

Depends on

Used by · 1

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, Section 59.69: Picard groups of curves · Opening paragraphs, before Lemma 59.69.1

For a smooth projective curve X of genus g over an algebraically closed field k, Pic⁰(X) is the group of k-points of the abelian variety Pic⁰_{X/k} of dimension g.

Hypotheses: X is a smooth projective curve over an algebraically closed field k; Pic⁰ denotes degree-zero line-bundle classes.

Reading limits: Read the displayed opening paragraphs and the identification stated there; the construction and proof of representability in the referenced Picard-schemes section were not reconstructed here.

Sources