Flexes and bitangents defined by intersection multiplicity

Let $C\subseteq {\mathbf{P}}^2$ be a plane projective curve over the algebraically closed field $k$ Plane projective curves and their components.

A smooth point $p\in C$ is a flex of $C$ when its tangent line $T_{p}C$ is not a component of $C$ and satisfies

$I_p(C,T_{p}C)\ge 3,$

and it is an ordinary flex when $I_p(C,T_{p}C)=3$. Here $T_{p}C$ is the unique tangent line at the smooth point $p$ Multiplicity one characterises smooth points with a unique tangent, Tangent cone and tangent lines at a point, and the contact is measured by the local intersection multiplicity of the curve with its tangent line Local intersection multiplicity of two plane curves.

A line $L$ not contained in $C$ is a bitangent of $C$ when $L$ has at least two distinct contact points of multiplicity at least two, that is, there are $p\mathrm{\ne }q$ in $L\cap C$ with $I_p(C,L)\ge 2$ and $I_q(C,L)\ge 2$. More generally, a line not contained in $C$ is a multitangent when the sum of the multiplicities at its contact points exceeds its number of contact points. No Plücker formula or duality statement is asserted.

The naming conventions above make no choice. The contact-order and degree-bound assertions below assume the Axiom of Choice inherited from their finite-length, DVR and Bezout suppliers The Axiom of Choice.

• Contact order. If $p$ is smooth and $L\subseteq ̸C$ is a line through $p$, then $I_p(C,L)={\mathrm{ord}}_p(F{\mathrm{\mid }}_L)$ for a local equation $F$ of $C$, the order of vanishing of the restricted equation along the line at $p$ Intersection with a line is the order of vanishing of the restricted equation. Equivalently $I_p(C,L)={\mathrm{ord}}_p(l{\mathrm{\mid }}_C)$ for a local equation $l$ of the line, with the valuation now taken along $C$ Intersection with a smooth curve is a vanishing order, Uniformising parameters at smooth points of a plane curve. So a flex is a point where the tangent line meets the curve with contact order at least three, and an ordinary flex is the case of contact order exactly three; a bitangent is a line whose contact with the curve has at least two double points.
• Finite contact. Smoothness does not prevent a tangent line from being a component: at a point of one line of a reducible curve away from the other components, the point is smooth and its tangent is that line. Such contacts have infinite intersection multiplicity and are excluded from the definitions above. In particular a line has no flexes. When $T_{p}C$ is not a component, the restriction of the defining form to it is nonzero and the local contact is finite.
• Degree bounds. A line $L\subseteq ̸C$ meets a degree-$d$ curve $C$ in exactly $d$ points counted with multiplicity, and hence at most $d$ distinct points A line meets a degree-d curve in d points counted with multiplicity, so flexes and bitangents of a degree-$d$ curve are subject to the classical counting constraints; the definition records the local data on which those counts rest without asserting any global formula.