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

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

Statement

The image of the Segre point map is precisely the nonzero projective matrices $(z_{i j})$ satisfying $z_{i j} z_{k ℓ} - z_{i ℓ} z_{k j} = 0 for all i, k, j, ℓ .$ It is a closed projective algebraic set. On every standard open $z_{i_{0} j_{0}} \neq 0$ , the two factors are recovered by regular coordinate ratios.

Facts & Assumptions

Given: A projective coordinate matrix $z = (z_{i j})$ .

Proof

1.1

If $z_{i j} = x_{i} y_{j}$ , then $z_{i j} z_{k ℓ} = x_{i} x_{k} y_{j} y_{ℓ} = z_{i ℓ} z_{k j}$ , so every Segre point satisfies every displayed minor.

givenalgebra

2.1

Conversely choose a nonzero entry $z_{i_{0} j_{0}}$ . The minor equations give $z_{i j} z_{i_{0} j_{0}} = z_{i j_{0}} z_{i_{0} j}$ , hence $z_{i j} = x_{i} y_{j}$ for $x_{i} = z_{i j_{0}}$ and $y_{j} = z_{i_{0} j} / z_{i_{0} j_{0}}$ . Thus the matrix is a Segre image.

step 1.1algebra

3.1

The minors are homogeneous quadrics, so their common zero locus is projectively closed, and step 2.1 identifies it with the image. On $z_{i_{0} j_{0}} \neq 0$ , the ratios $\frac{z_{i j_{0}}}{z_{i_{0} j_{0}}}, \frac{z_{i_{0} j}}{z_{i_{0} j_{0}}}$ are regular and recover the standard affine coordinates of both factors. Thus the asserted chartwise inverse is proved directly; injectivity alone is not used as an embedding criterion.

step 2.1algebra∎

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Dependency tree · two levels

9 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, 6.26 (standard reference, not scraped)
MIT 18.725 Algebraic Geometry, Lecture 7, The Segre embedding (standard reference, not scraped)