LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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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.

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The Segre map is well defined and injective

Statement

The Segre point map is a well-defined injective map of point sets. This includes $m = 0$ or $n = 0$ .

Facts & Assumptions

Given: Representatives $x \neq 0$ and $y \neq 0$ .

Proof

1.1

Replacing $x, y$ by $λ x, μ y$ replaces every coordinate $x_{i} y_{j}$ by the single nonzero scalar $λ μ$ times it. Some product $x_{i_{0}} y_{j_{0}}$ is nonzero, so these coordinates define a well-defined projective point.

givenalgebra

2.1

Choose $i_{0}, j_{0}$ with $x_{i_{0}} \neq 0$ and $y_{j_{0}} \neq 0$ . From a projective matrix $[z_{i j}]$ in the image, the ratios $z_{i j_{0}} : z_{i_{0} j_{0}}$ recover $[x_{i}]$ , and $z_{i_{0} j} : z_{i_{0} j_{0}}$ recover $[y_{j}]$ .

step 1.1algebra

3.1

Thus equal Segre images have equal two projective factors. The same calculation works when one factor has one homogeneous coordinate, proving the endpoint cases.

step 2.1∎

Depends on

The Segre point map from two projective spaces

Used by

The Segre image is the projective rank-one locus cut out by 2 by 2 minors Theorem

Dependency tree · two levels

4 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

J. S. Milne, Algebraic Geometry, The Segre map, §6i (standard reference, not scraped)