CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Dimension is birationally invariant

Statement

Birational irreducible classical varieties have equal dimension. In particular, if $X$ and $Y$ are irreducible classical varieties, $U \subseteq X$ and $V \subseteq Y$ are nonempty open subvarieties, and $U ≅ V$ , then $\dim X = \dim Y$ .

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]

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).

[F2]

Under AC, birational equivalence of integral classical varieties means the existence of isomorphic nonempty open subsets (Birational maps and birational equivalence of classical varieties).

Proof

1.1F1

An isomorphism $U ≅ V$ carries irreducible closed chains to chains of the same length. Open invariance gives $\dim X = \dim U = \dim V = \dim Y$ .

2.1F2step 1.1∎

For birational irreducible classical varieties, [F2] supplies isomorphic nonempty opens. Step 1.1 applies to them and proves equality of dimensions.

Depends on

Used by

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Dependency tree · two levels

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Sources

Milne §5l, Proposition 5.39, p.117 (standard reference, not scraped)