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

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.

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 $i$ .

1.1givenalgebra

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

2.1step 1.1

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

3.1step 2.1algebra∎

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

Depends on

Used by

Nontrivial degree-zero line bundles have no sections Corollary
The genus of a smooth plane curve in terms of its degree Corollary
The Riemann inequality is not an equality for special divisors Counterexample
Multiplicity of a plane curve at a point Definition
Plane projective curves and their components Definition
A nondegenerate projective quadric Example
A projective cone with a smooth conic base Example
projective conic standard charts Example
Smooth plane quartic has genus three Example
Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
Multiplicity one characterises smooth points with a unique tangent Lemma
Adjunction for smooth plane curves Theorem

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)