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.

Elementary fibres: empty, points, and affine lines

Example

For $H = V (x y - 1) \subseteq A_{k}^{2}$ , projection $H \to A_{k}^{1}$ onto $x$ has fibre ${(a, a^{- 1})}$ when $a \neq 0$ and empty fibre at $a = 0$ . Their dimensions are zero and $- \infty$ respectively. By contrast, every fibre of $A^{2} \to A^{1}$ , $(x, y) \mapsto x$ , is an affine line of dimension one.

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]

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

Fixing $x = a$ in $x y = 1$ gives $a y = 1$ . If $a \neq 0$ the unique solution is $y = a^{- 1}$ ; if $a = 0$ it has no solution. These reduced fibres are respectively a point and the empty variety, of dimensions zero and $- \infty$ .

F1F2

2.1

For the full-plane projection fixing $x = a$ leaves $y$ arbitrary. The regular maps $y \mapsto (a, y)$ and $(a, y) \mapsto y$ are inverse isomorphisms of the fibre with $A^{1}$ . Thus its dimension is one for every $a$ , including zero.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Milne Example 8.30, p.185; full-plane comparison computed here (standard reference, not scraped)