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The component-counting obstruction template for incidence arguments

Statement

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

Let $C$ be a plane projective curve of degree $d$ and $E$ a plane projective curve of degree $f$ over the algebraically closed field $k$ . If $C$ and $E$ have no common component then $E$ contains at most $d f$ points of $C$ , and any set of pairwise distinct points of $C$ on which $E$ is required to vanish must have at most $d f$ elements. Consequently, in any incidence configuration in which curves of degrees $d$ and $f$ are forced to share more than $d f$ distinct points, the two curves must share a component. This is the standard component-counting step behind Pascal- and Pappus-type applications, isolated here without minting a separate named incidence theorem.

Facts & Assumptions

Given: AC The Axiom of Choice, plane projective curves $C$ of degree $d \geq 1$ and $E$ of degree $f \geq 1$ over the algebraically closed field $k$ .

[F1]

If $C, E$ have no common component, then $\sum_{p \in C \cap E} I_{p} (C, E) = d f$ over the finitely many intersection points Bezout's theorem for plane projective curves.

[F2]

Whenever $I_{p} (C, E)$ is finite, it is a positive integer at points of $C \cap E$ and zero at points outside the intersection Symmetry, additivity and local nature of intersection multiplicity. Under the no-common-component hypothesis, [F1] ensures this finiteness at every intersection point.

Proof

1.1F1F2algebra

Assume $C$ and $E$ have no common component. By [F2] every point of $C \cap E$ contributes at least one to the Bezout sum, so $# (C \cap E) \leq \sum_{p \in C \cap E} I_{p} (C, E) = d f$ ; in particular $E$ contains at most $d f$ points of $C$ .

2.1step 1.1given

If $S$ is a set of pairwise distinct points of $C$ at which $E$ is required to vanish, then $S \subseteq C \cap E$ , so by step 1.1 $∣ S ∣ \leq d f$ .

3.1step 1.1step 2.1F1∎

Consequently, if an incidence configuration forces more than $d f$ distinct common points, the assumption of no common component is impossible, so $C$ and $E$ share a component; this is the reusable obstruction template, and the finiteness and nonemptiness statements accompanying it are [F1] and Two plane projective curves meet.

Depends on

Used by

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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)