intersection divisor for a quartic

Let $C$ be a non-singular 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 $1\leq i\neq j\leq 4$ such that $P_{i}=P_{j}$ . Without loss of generality we may assume $P_{1}=P_{2}$ and $(L\cdot C)=2P_{1}+P_{3}+P_{4}$ , and $P_{3}\neq P_{4}$ .
3.

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

$L$ intersects $C$ at exactly two points, thus $P_{1}=P_{2}=P_{3}\neq P_{4}$ . 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 non-hyperelliptic 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	2013-03-22 15:44:59
Last modified on	2013-03-22 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