DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Dominant morphisms and dominant rational maps

Definition

Let $X$ and $Y$ be classical affine varieties.

A morphism $φ : X \to Y$ is dominant when its image $φ (X)$ is dense in $Y$ .

A rational map $η : X ⇢ Y$ is dominant when every representative morphism $φ : U \to Y$ has dense image in $Y$ .

The representatives are compatible on their overlaps and regularity is local, so they glue to a unique regular map

η_{\max} : Dom (η) ⟶ Y .

This is the maximal representative of $η$ . In the dominant case its image is dense in $Y$ , since it contains the image of every representative.

Depends on

Used by

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, Proposition 3.34(a) and Chapter 5k (standard reference, not scraped)
J. S. Milne, Algebraic Geometry, Proposition 5.38 (standard reference, not scraped)