LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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projective hypersurface affine pieces

Statement

If $X = V_{+} (F)$ , then $X \cap D_{+} (x_{i})$ is the affine hypersurface obtained by setting $x_{i} = 1$ in $F$ , with the usual ratio-coordinate transition formulas.

Proof

Given: A homogeneous polynomial $F$ and an index $i$ .

1.1

Normalize a point of $D_{+} (x_{i})$ by $x_{i} = 1$ ; $F$ vanishes exactly when the dehomogenized equation vanishes.

givenalgebra

2.1

The chart normalization therefore identifies $X \cap D_{+} (x_{i})$ with that affine hypersurface.

step 1.1

3.1

On overlaps the two normalizations differ by division by a nonzero coordinate, giving the stated transitions.

step 2.1algebra∎

Depends on

Used by

projective conic standard charts Example

Dependency tree · two levels

7 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, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)