Alphabeta MathHow 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.
Normalization need not resolve singularities in dimension at least two
Remarks
Assume the Axiom of Choice, as in the curve suppliers (The Axiom of Choice).
For curves over a perfect field, normalization removes all singularities: A normal curve over a perfect field is nonsingular shows that a normal curve is nonsingular, For projective curves, Normalization resolves the singularities of a projective curve constructs the finite projective normalization over the actual perfect field.
In dimension at least two this fails. A normal variety can still be singular: the quadric cone over an algebraically closed field of characteristic not two is normal with an isolated singular point, as the companion counterexample A normal singular surface: the quadric cone ↗ shows. Since that cone is already normal, its normalization is the identity by The normalization of an irreducible affine variety and does not remove the singularity.
Normalization therefore only removes the non-normal locus, not the singular locus in general, and resolution of singularities in dimension at least two is a separate subject. This remark records the boundary of the curve theorem and asserts no resolution statement.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
37 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.