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

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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

1.1givenalgebra

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.

2.1step 1.1algebra

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 $[y_{j}]$ .

3.1step 2.1∎

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

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

2 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)