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

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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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Symmetry, additivity and local nature of intersection multiplicity

Statement

Assume the Axiom of Choice. Let $C, D$ be plane projective curves over the algebraically closed field $k$ and let $p \in P^{2}$ , with all local intersection multiplicities below assumed finite. Then:

$I_{p} (C, D) = I_{p} (D, C)$ .
$I_{p} (C, D) = 0$ iff $p \notin C \cap D$ , and $I_{p} (C, D) \geq 1$ iff $p \in C \cap D$ .
If $G_{1}, G_{2}$ are square-free forms with no common factor, so that $V (G_{1} G_{2})$ is again a plane projective curve, and if the three values are finite, then $I_{p} (C, V (G_{1} G_{2})) = I_{p} (C, V (G_{1})) + I_{p} (C, V (G_{2})) .$
If $G^{'}$ is a form such that the local equation of $V (G^{'})$ at $p$ differs from that of $D$ by a multiple of a local equation of $C$ — in particular if $G^{'} = G + A F$ with $A$ a form of degree $\deg G - \deg F$ and $V (G^{'})$ is a plane curve — then $I_{p} (C, V (G^{'})) = I_{p} (C, D)$ .
$I_{p} (C, D)$ depends only on the local branches of $C$ and $D$ through $p$ .

Facts & Assumptions

Given: AC, plane curves $C = V (F)$ , $D = V (G)$ over the algebraically closed field $k$ , a point $p$ , a common local ring $O = O_{P^{2}, p}$ with maximal ideal $m_{p}$ , and local equations $f, g$ of $C, D$ at $p$ ; $I_{p} (C, D) = ℓ_{O} (O / (f, g))$ Local intersection multiplicity of two plane curves.

[F1]

$O$ is the localisation of the polynomial UFD of the plane, hence a unique factorisation domain by the explicit factorisation argument in the finiteness lemma (Proof 1.3); two local equations have a common irreducible factor exactly when the curves share a local branch at $p$ , and then the length is infinite, while $O / (f, g)$ has finite length exactly when $f, g$ are coprime in $O$ Finite local length exactly when no common local branch, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Unique factorisation domain, Irreducible and prime elements of an integral domain.

[F2]

Length is additive in short exact sequences of $O$ -modules, and the length of a module depends only on its isomorphism class Module length is additive in short exact sequences, Composition series and length of a module. Kernels and images of module homomorphisms and quotients by ideals are the usual ones Module homomorphism and isomorphism, kernel, image and cokernel, The quotient ring $R / I$ with $(r + I) (s + I) = r s + I$ .

[F3]

$I_{p}$ is unchanged by replacing local equations by other generators of the same local ideal, by chart changes and by affine or projective coordinate changes Invariance of the local intersection multiplicity, Localising twice is localising once at the multiplicative set generated by both denominator sets, Localisation commutes with quotient rings: $S^{- 1} R / S^{- 1} I ≅ {\overset{ˉ}{S}}^{- 1} (R / I)$ .

[F4]

AC is assumed; it enters through the finiteness and localisation suppliers The Axiom of Choice.

Proof

1.1F2given

Symmetry: $O / (f, g) = O / (g, f)$ , so $I_{p} (C, D) = I_{p} (D, C)$ .

1.2F2givenalgebra

Vanishing: if $p \notin C \cap D$ , one of $f, g$ is a unit of $O$ , so $(f, g) = O$ and $O / (f, g) = 0$ has length $0$ ; if $p \in C \cap D$ , then $f, g \in m_{p}$ , so $(f, g) \subseteq m_{p}$ and the quotient $O / (f, g)$ maps onto $O / m_{p} \neq 0$ , hence has positive length.

1.3F1F2algebra

Additivity: with $G_{1}, G_{2}$ square-free and coprime and all values finite, let $g_{2}$ be the local equation of $V (G_{2})$ . Since $I_{p} (C, V (G_{2}))$ is finite, [F1] shows $f$ and $g_{2}$ are coprime in $O$ . Consider the sequence of $O$ -modules $0 ⟶ O / (f, g_{1}) \overset{\cdot g_{2}}{\to} O / (f, g_{1} g_{2}) \overset{π}{\to} O / (f, g_{2}) ⟶ 0.$ The map $π$ is the natural reduction, and its kernel is $(f, g_{2}) / (f, g_{1} g_{2})$ , which is exactly the image of multiplication by $g_{2}$ , so the sequence is exact at the middle; injectivity of multiplication by $g_{2}$ holds because $g_{2} z \in (f, g_{1} g_{2})$ implies $f ∣ g_{2} (z - b g_{1})$ for some $b$ , and coprimality of $f$ and $g_{2}$ gives $z \in (f, g_{1})$ . Length additivity in [F2] now gives the displayed identity.

1.4F2F3givenalgebra

Invariance under adding a multiple: if $G^{'}$ has local equation $g^{'} = g + a f$ at $p$ , then $(f, g^{'}) = (f, g)$ as ideals of $O$ , so $I_{p} (C, V (G^{'})) = ℓ_{O} (O / (f, g^{'})) = ℓ_{O} (O / (f, g)) = I_{p} (C, D)$ ; for a form $G^{'} = G + A F$ of the same degree as $G$ the local equation has exactly this shape by [F3].

1.5F1F3given

Locality: the quotient $O / (f, g)$ is computed from the germs of local equations at $p$ , so replacing $C, D$ by curves with the same local branches at $p$ leaves $f, g$ unchanged up to units in $O$ and leaves the length unchanged; this is the content of [F3].

2.1step 1.1step 1.2step 1.3step 1.4step 1.5F4∎

Statements (1)–(5) are established in steps 1.1, 1.2, 1.3, 1.4 and 1.5.

Depends on

Used by

A line meets a degree-d curve in d points counted with multiplicity Corollary
The component-counting obstruction template for incidence arguments Corollary
Two plane projective curves meet Corollary
Bezout fails on the affine plane because points at infinity are missing Counterexample
Bezout needs algebraic closure: an imaginary conic has no real point Counterexample
Curves sharing too many points share a component Theorem

Dependency tree · two levels

81 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)
MIT 18.725 Algebraic Geometry (Fall 2015) lecture notes, consolidated (standard reference, not scraped)