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

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.

Standard affine charts on the Grassmannian

Statement

For every $r$ -subset $I$ of a basis of an $n$ -dimensional $V$ , the locus $p_{I} \neq 0$ in $Gr (r, V)$ is isomorphic to $A_{k}^{r (n - r)}$ ; these loci cover the Grassmannian and have regular transition maps.

Proof

Given: A plane $S$ with nonzero Plucker coordinate $p_{I}$ .

1.1

Reorder the basis so $I = {1, \dots, r}$ . The projection $S \to k^{r}$ is invertible, so $S$ has the unique row-space representative $(I_{r} ∣ A)$ with $A \in {Mat}_{r, n - r} (k)$ .

givenalgebra

2.1

The $r (n - r)$ entries of $A$ give affine coordinates, and every Plucker coordinate is a minor of $(I_{r} ∣ A)$ , hence a polynomial in them. Conversely these entries are ratios of Plucker coordinates by $p_{I}$ .

step 1.1algebra

3.1

Thus this locus is affine space. On an overlap, changing the pivot columns replaces $(I_{r} ∣ A)$ by multiplication with the inverse of an invertible minor, so the transition entries are regular ratios on that overlap.

step 2.1∎

Depends on

Used by

The Grassmannian is smooth, irreducible, and has dimension r(n-r) Corollary

Dependency tree · two levels

8 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, Remark 6.31 (standard reference, not scraped)