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. 1.

The generic position: all the points $P_{i}$ are distinct.

2. 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. 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. 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. 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 IntersectionDivisorForAQuartic 2013-03-22 15:44:59 2013-03-22 15:44:59 alozano (2414) alozano (2414) 5 alozano (2414) Definition msc 14C20 hyperflex flex