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Flexes are contacts of order at least three with the tangent line

Statement

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

Let $C$ be a plane projective curve over the algebraically closed field $k$ and let $p \in C$ be a smooth point with tangent line $T = T_{p} C$ . Then $p$ is a flex of $C$ if and only if $T$ is not a component of $C$ and the nonzero restriction to $T$ of a defining form of $C$ vanishes at $p$ to order at least three. Whenever $T$ is not a component,

$I_{p} (C, T) = {ord}_{p} (F ∣_{T}) .$

In particular $p$ is an ordinary flex exactly when $I_{p} (C, T) = 3$ , and for a smooth conic or a line there are no flexes.

Facts & Assumptions

Given: AC The Axiom of Choice, a plane projective curve $C = V (F)$ of degree $d$ over the algebraically closed field $k$ and a smooth point $p \in C$ with tangent line $T = T_{p} C$ Plane projective curves and their components, Multiplicity one characterises smooth points with a unique tangent.

[F1]

A flex is a smooth point whose tangent line is not a component and has $I_{p} (C, T_{p} C) \geq 3$ , an ordinary flex one with $I_{p} (C, T_{p} C) = 3$ Flexes and bitangents defined by intersection multiplicity.

[F2]

If the tangent line $T$ is not a component of $C$ , then $I_{p} (C, T) = {ord}_{p} (F ∣_{T})$ , with order taken in the DVR $O_{T, p}$ , by Intersection with a line is the order of vanishing of the restricted equation. The equivalent valuation along the smooth curve $C$ is ${ord}_{p} (l ∣_{C})$ for a local equation $l$ of $T$ Intersection with a smooth curve is a vanishing order, Uniformising parameters at smooth points of a plane curve.

[F3]

When $T$ is not contained in $C$ , the restriction $F ∣_{T}$ is a nonzero binary form of degree $d$ ; the order of vanishing at $p$ is at most $d$ by the root bound for the restriction Flexes and bitangents defined by intersection multiplicity, Plane projective curves and their components.

Proof

1.1F1F2F3given

If $T$ is a component of $C$ , its contact has infinite multiplicity and $p$ is not a flex by [F1]. Otherwise regard $T$ as the smooth curve and restrict the local equation $F$ of $C$ to it. By symmetry of the defining local quotient and [F2], $I_{p} (C, T) = {ord}_{p} (F ∣_{T})$ ; [F3] makes this finite and at most $d$ .

2.1F1step 1.1algebra

In the finite-contact case of step 1.1, by [F1] the point $p$ is a flex exactly when $I_{p} (C, T) \geq 3$ , i.e. by step 1.1 exactly when ${ord}_{p} (F ∣_{T}) \geq 3$ ; it is an ordinary flex exactly when both are $3$ .

2.2step 1.1F1F3given

If $C$ is a smooth conic and its tangent is not a component, then $d = 2$ , so ${ord}_{p} (F ∣_{T}) \leq 2 < 3$ and no point is a flex. If $C$ is a line, then $T_{p} C = C$ for every point, the pair $(C, T_{p} C)$ has a common component and $I_{p}$ is not finite, so no point is a flex in the sense of the definition.

3.1step 1.1step 2.1step 2.2∎

Steps 1.1, 2.1 and 2.2 establish the identification, the characterisation of ordinary flexes by contact order three, and the absence of flexes on smooth conics and lines.

Depends on

Used by

A flex of a cubic has contact order three Example

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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)