CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-6.1-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.

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Counting distinct points is not enough: tangent contact

Statement refuted

False claim: for two plane projective curves of degrees $d, e$ over an algebraically closed field, the number of distinct intersection points equals $d e$ .

Facts & Assumptions

Given: AC The Axiom of Choice, the conic $C = V (x_{0}^{2} - x_{1} x_{2})$ and its tangent line $L = V (x_{1})$ at $p = [0 : 0 : 1]$ , both over an algebraically closed field of characteristic not two.

[F1]

$L \cap C = {p}$ as a set, and $I_{p} (C, L) = 2$ : the restriction of the conic equation to $L$ is $x_{0}^{2}$ , a double root at $p$ A tangent line meets a conic with multiplicity two at one point.

[F2]

Bezout for $\deg C = 2$ , $\deg L = 1$ gives $\sum_{q \in C \cap L} I_{q} (C, L) = 2$ over an algebraically closed field Bezout's theorem for plane projective curves, and every local multiplicity at a point of the intersection is a positive integer Local intersection multiplicity of two plane curves.

Counterexample

1.1F1given

The distinct intersection points number one, while the degree product is $d e = 2 \cdot 1 = 2$ .

1.2F1F2given

The multiplicity-weighted sum is $I_{p} (C, L) = 2$ , the value required by Bezout, concentrated at the unique point.

2.1step 1.1step 1.2F1F2∎

Therefore the distinct-point count $1$ differs from $d e = 2$ ; the deficiency is repaired exactly by counting the tangent contact with multiplicity two, so the number of distinct points alone does not equal the degree product.

Depends on

Used by

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Dependency tree · two levels

23 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)