CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-sol)

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.

Transversal smooth curves meet with multiplicity one

Statement

Assume the Axiom of Choice, inherited from the cited local-length, smoothness or Bezout suppliers.

Let $C, D$ be plane projective curves over the algebraically closed field $k$ meeting at $p$ transversally: both are smooth at $p$ and their tangent lines at $p$ are distinct. Then $m_{p} (C) = m_{p} (D) = 1$ and $I_{p} (C, D) = 1$ . Conversely $I_{p} (C, D) = 1$ forces $C$ and $D$ to be smooth at $p$ with distinct tangent lines.

Facts & Assumptions

Given: AC The Axiom of Choice, plane projective curves $C, D$ over the algebraically closed field $k$ meeting at $p$ , with $C, D$ smooth at $p$ and distinct tangent lines.

[F1]

A point of a plane curve is smooth exactly when its multiplicity is one, and then the curve has exactly one tangent line at that point Multiplicity one characterises smooth points with a unique tangent, Tangent cone and tangent lines at a point, Multiplicity of a plane curve at a point.

[F2]

If $C, D$ have no common local component at $p$ , then $I_{p} (C, D) \geq m_{p} (C) m_{p} (D)$ , with equality exactly when the tangent cones share no line Intersection multiplicity dominates the product of multiplicities, with equality for separated tangent cones, Local intersection multiplicity of two plane curves. Smooth curves at a common point with distinct tangent lines have no common local component there, since a common branch would force a common tangent line.

[F3]

At a point where they share no local component, $I_{p} (C, D) \geq 1$ exactly when $p \in C \cap D$ Local intersection multiplicity of two plane curves.

Proof

1.1F1F2given

By [F1] smoothness at $p$ gives $m_{p} (C) = m_{p} (D) = 1$ ; distinct tangent lines mean the tangent cones share no line, and then the two curves share no local component at $p$ . Applying the product inequality [F2] with $m = n = 1$ gives $I_{p} (C, D) \geq 1$ , and since the tangent cones are separated, equality holds: $I_{p} (C, D) = 1$ .

1.2F1F2algebraF3

Conversely suppose $I_{p} (C, D) = 1$ . Since the value is finite, $C$ and $D$ share no local component at $p$ , so [F2] applies with $m = m_{p} (C) \geq 1$ , $n = m_{p} (D) \geq 1$ : $1 = I_{p} (C, D) \geq m n \geq 1$ , hence $m n = 1$ , so $m = n = 1$ , and equality in the product inequality holds; therefore by [F2] the tangent cones share no line. By [F1], $m_{p} (C) = m_{p} (D) = 1$ means both curves are smooth at $p$ , each with a unique tangent line, and the tangent lines are distinct.

2.1step 1.1step 1.2∎

The two implications establish the equivalence: transversal smooth curves have local multiplicity one, and local multiplicity one forces smoothness with distinct tangents.

Depends on

Used by

Bezout needs algebraic closure: an imaginary conic has no real point Counterexample
A line and a conic meet in two points counted with multiplicity Example
Two transverse cubics meet in nine points Example

Dependency tree · two levels

64 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

William Fulton, Algebraic Curves: An Introduction to Algebraic Geometry (2008 electronic edition; Internet Archive copy of the author's PDF) (standard reference, not scraped)
Michael Artin, MIT 18.721 Notes for a Course in Algebraic Geometry (January 26, 2022 version), Chapter 1 (standard reference, not scraped)