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

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The Plucker map is well defined and injective

Statement

The Plucker map $pl : Gr (r, V) \to P (Λ^{r} V)$ is well defined and injective.

Proof

Given: An $r$ -plane $S$ with ordered basis $(v_{1}, \dots, v_{r})$ .

1.1givenalgebra

A change of basis multiplies $v_{1} \land \dots \land v_{r}$ by its nonzero determinant, so its projective line is independent of the chosen basis. The wedge is nonzero because the basis is independent.

2.1step 1.1algebra

Let $w = v_{1} \land \dots \land v_{r}$ . A vector $u$ satisfies $u \land w = 0$ exactly when $u \in S$ : one direction has a repeated vector, and the other follows because $u, v_{1}, \dots, v_{r}$ are independent when $u \notin S$ , so their wedge is nonzero.

3.1step 2.1∎

This annihilator description recovers $S$ from the projective line $[w]$ . Equal Plucker points therefore determine equal subspaces.

Depends on

Used by

Dependency tree · two levels

11 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, Proposition 6.29 and Remark 6.34 (standard reference, not scraped)