LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)audited 2026-10-02

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The projective-line twisting sheaf is ample

Statement

Assume the Axiom of Choice. The sheaf $O (1)$ on $P_{k}^{1}$ is ample for every field $k$ .

Facts & Assumptions

Given: AC, a field $k$ and $P_{k}^{1}$ with its standard twisting sheaf.

[F1]

The structure morphism of projective space is finite type, hence quasi-compact; the standard twist is invertible. (Projective space is of finite type over its base, Locally finite type and finite type morphisms, Relative very ampleness in the finite projective-space convention)

[F2]

Under AC, H-very ampleness for a quasi-compact morphism implies relative ampleness, and implies absolute ampleness over an affine base. (Relative very ampleness implies relative ampleness, The Axiom of Choice)

Proof

1.1F1construct

The identity of $P_{k}^{1}$ is a quasi-compact closed immersion over $Spec k$ and pulls $O (1)$ back to itself. It therefore witnesses closed H-very ampleness of the standard twist.

2.1F1F2step 1.1

Since $Spec k$ is affine, [F2] implies absolute ampleness. AC is used only through the cited projective-space and very-ample-to-ample suppliers. ∎

Depends on

Used by

Ramification of the double cover y²=f(x) Example

Dependency tree · two levels

26 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

The Stacks Project; elementary local prerequisite for the Step 5b citation repair (standard reference, not scraped)