ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-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.

The map (x,y) to (x,xy) has a jumping fibre

Example

The morphism $f : A_{k}^{2} \to A_{k}^{2}$ , $(x, y) \mapsto (u, v) = (x, x y)$ , has image $D (u) \cup {(0, 0)}$ . Its fibres are one point when $u \neq 0$ , an affine line over $(0, 0)$ , and empty over $(0, v)$ with $v \neq 0$ . The image is constructible and is not locally closed.

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]

For a morphism $f : X \to Y$ of classical varieties and a closed point $y \in Y$ , let $X_{y} = f^{- 1} (y)$ have its reduced closed-subvariety structure. Its dimension is the chain dimension, with $\dim X_{y} = - \infty$ if the fibre is empty. On affine charts $V \subseteq Y$ containing $y$ and $U \subseteq f^{- 1} (V)$ , writing $A = k [V]$ and $B = k [U]$ , the fibre chart has coordinate ring $B / \sqrt{m_{y} B}$ . Here general morphisms have the locally ringed-space meaning; the earlier affine morphism definition applies to the restrictions $U \to V$ . 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. (Reduced closed-point fibres and their dimension).

[F2]

A subset $S$ of a classical variety $X$ is locally closed if $S = U \cap Z$ for some open $U \subseteq X$ and closed $Z \subseteq X$ . A subset is constructible if it is a finite union of locally closed subsets. The empty union is allowed, so $\emptyset$ is constructible. 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. (Locally closed and constructible subsets).

[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).

Verification

1.1

The fibre equations are $x = u$ and $u y = v$ . For $u \neq 0$ there is exactly one solution $(u, v / u)$ , of dimension zero. If $u = v = 0$ , $y$ is free, so the fibre is an affine line of dimension one. If $u = 0$ and $v \neq 0$ there is no solution. This proves the image description and the empty-fibre dimension $- \infty$ .

F1F3

2.1

The image is a union of an open subset and a closed point, hence constructible. It is dense since $D (u)$ is dense in the irreducible affine plane. If this image were locally closed, writing it as open intersect closed and taking closure would show it open in the plane. But any open neighborhood of the origin meets the line $u = 0$ in a nonempty open subset of that line. A proper closed subset of an affine line is finite by the polynomial root bound, so this neighborhood contains a point $(0, v)$ with $v \neq 0$ , outside the image. Therefore the image is not open and not locally closed.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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 Chapter 9 introduction, p.198 (standard reference, not scraped)
Arapura Example 4.2.4, p.32 (standard reference, not scraped)