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.

Images and fibres of a regular map

Definition

Let $φ : X \to Y$ be a morphism of classical affine varieties.

The image of $φ$ is the subset $φ (X) : = {φ (x) : x \in X} \subseteq Y .$

For a point $y \in Y$ , the fibre of $φ$ over $y$ is $φ^{- 1} (y) : = {x \in X : φ (x) = y} .$

More generally, for a subset $S \subseteq Y$ one writes $φ^{- 1} (S) : = {x \in X : φ (x) \in S} .$

These are set-theoretic notions. On this page no separate claim is made that images are closed.

Depends on

Morphisms of classical affine varieties

Used by

Normalization of the node is two-to-one over the node Counterexample
Unibranch points of a classical variety Definition

Dependency tree · two levels

3 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

Michael Artin, Notes for a Course in Algebraic Geometry, after Proposition 2.5.8 (standard reference, not scraped)
J. S. Milne, Algebraic Geometry, Chapter 3h (standard reference, not scraped)