ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge 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.

Plane curves meet; common components change the dimension

Example

Two distinct lines in $P_{k}^{2}$ intersect in one point. The reducible curves $V_{+} (X Y)$ and $V_{+} (X Z)$ intersect in the line $X = 0$ together with the point $[1 : 0 : 0]$ . In contrast, two distinct irreducible projective plane curves have a nonempty finite intersection.

Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points.

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

Let $X, Y \subseteq P_{k}^{n}$ be irreducible closed subvarieties. Every nonempty irreducible component $Z$ of $X \cap Y$ satisfies $\dim Z \geq \dim X + \dim Y - n$ . If $\dim X + \dim Y \geq n$ , then $X \cap Y \neq \emptyset$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Projective intersection dimension and nonemptiness).

[F2]

If a Noetherian space $T$ is a finite union of closed subsets $T_{1}, \dots, T_{m}$ , then $\dim T = \max_{i} \dim T_{i}$ . For $m = 0$ both sides are $- \infty$ . (Dimension of a finite closed union).

[F3]

For every integer $n \geq 0$ , $\dim A_{k}^{n} = \dim P_{k}^{n} = n$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Affine and projective n-space have dimension n).

[F4]

If $U$ is a nonempty open of an irreducible classical variety $X$ , then $\dim U = \dim X$ . Every proper closed subvariety $Z ⊊ X$ has $\dim Z < \dim X$ . Work over a fixed algebraically closed field $k$ , with the Axiom of Choice. Classical varieties are separated and admit finite affine covers; they may be reducible or empty unless irreducibility is specified. Irreducible means nonempty. All fibres and points below are classical closed-point fibres and points. (Nonempty opens preserve irreducible dimension).

Verification

1.1

Two distinct lines are defined by independent linear forms on $k^{3}$ . Their common kernel has vector dimension one, so its projectivization is a single point. For the displayed reducible curves, the equations $X Y = X Z = 0$ imply either $X = 0$ , giving the whole line, or $X \neq 0$ and $Y = Z = 0$ , giving $[1 : 0 : 0]$ . This point is outside the line. The intersection has dimension one, as a finite closed union of a line and a point.

F2F3

2.1

For distinct irreducible plane curves $C, D$ of dimension one, the projective intersection theorem ensures nonemptiness because $1 + 1 = 2$ . Their intersection is a proper closed subset of $C$ : otherwise $C \subseteq D$ , and a proper closed subset of irreducible $D$ could not have dimension one. Thus all components of $C \cap D$ have dimension zero by proper-closed dimension drop. There are finitely many components, each a point (a larger irreducible closed set would contain a singleton chain of length one). Consequently the intersection is finite.

F1F4step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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

Milne Corollary 6.47, p.156; explicit line and reducible-curve computations supplied here (standard reference, not scraped)