intersection divisor for a quartic

Let C be a non-singularPlanetmathPlanetmath 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 K2. The intersectionMathworldPlanetmath divisor of C and L is of the form:


where Pi, i=1,2,3,4, are points in C(K). There are five possibilities:

  1. 1.

    The generic position: all the points Pi are distinct.

  2. 2.

    L is tangent to C: there exist indices 1ij4 such that Pi=Pj. Without loss of generality we may assume P1=P2 and (LC)=2P1+P3+P4, and P3P4.

  3. 3.

    L is bitangent to C when P1=P2 and P3=P4 but P1P3. It may be shown that if char(K)2 then C has exactly 28 bitangent lines.

  4. 4.

    L intersects C at exactly two points, thus P1=P2=P3P4. The point P1 is called a flex.

  5. 5.

    L intersects C at exactly one point and P1=P2=P3=P4. This point is called a hyperflex. A quartic C may not have any hyperflex.


  • 1 S. Flon, R. Oyono, C. Ritzenthaler, Fast additionPlanetmathPlanetmath on non-hyperelliptic genus 3 curves, can be found
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