intersection divisor for a quartic
Let $C$ be a nonsingular^{} curve in the plane, defined over an algebraically closed field $K$, and given by a polynomial $f(x,y)=0$ of degree $4$ (i.e. $C$ is a quartic). Let $L$ be a (rational) line in the plane ${K}^{2}$. The intersection^{} divisor of $C$ and $L$ is of the form:
$$(L\cdot C)={P}_{1}+{P}_{2}+{P}_{3}+{P}_{4}$$ 
where ${P}_{i}$, $i=1,2,3,4$, are points in $C(K)$. There are five possibilities:

1.
The generic position: all the points ${P}_{i}$ are distinct.

2.
$L$ is tangent to $C$: there exist indices $1\le i\ne j\le 4$ such that ${P}_{i}={P}_{j}$. Without loss of generality we may assume ${P}_{1}={P}_{2}$ and $(L\cdot C)=2{P}_{1}+{P}_{3}+{P}_{4}$, and ${P}_{3}\ne {P}_{4}$.

3.
$L$ is bitangent to $C$ when ${P}_{1}={P}_{2}$ and ${P}_{3}={P}_{4}$ but ${P}_{1}\ne {P}_{3}$. It may be shown that if $\mathrm{char}(K)\ne 2$ then $C$ has exactly $28$ bitangent lines.

4.
$L$ intersects $C$ at exactly two points, thus ${P}_{1}={P}_{2}={P}_{3}\ne {P}_{4}$. The point ${P}_{1}$ is called a flex.

5.
$L$ intersects $C$ at exactly one point and ${P}_{1}={P}_{2}={P}_{3}={P}_{4}$. This point is called a hyperflex. A quartic $C$ may not have any hyperflex.
References
 1 S. Flon, R. Oyono, C. Ritzenthaler, Fast addition^{} on nonhyperelliptic genus 3 curves, can be found http://eprint.iacr.org/2004/118.pshere.
Title  intersection divisor for a quartic 

Canonical name  IntersectionDivisorForAQuartic 
Date of creation  20130322 15:44:59 
Last modified on  20130322 15:44:59 
Owner  alozano (2414) 
Last modified by  alozano (2414) 
Numerical id  5 
Author  alozano (2414) 
Entry type  Definition 
Classification  msc 14C20 
Defines  hyperflex 
Defines  flex 