CU-27P1 The 27 Sextatic Points of a Cubic Curve
The 27 Sextactic Points on CU
In classical algebraic geometry, a sextactic point on a plane curve is a point where the curve has unusually high contact with a conic.
The study of sextactic points dates back to 19th-century projective geometry, notably in the work of Cayley, Salmon, and Clebsch.
See also [77], page 137 (option 13.) and [80].
For a smooth irreducible cubic curve, the sextactic points are defined as follows:
Definition
A point P on CU is called a sextactic point if there exists a conic Ci such that:
- Q osculates CU at P to order six, i.e., the intersection multiplicity ≥6.
- This contact is exceeds the generic case, since a general conic intersects a cubic curve in 6 points (by Bézout’s theorem), but not all at one point.
Properties
- A smooth cubic curve has exactly 27 distinct sextactic points over an algebraically closed field of characteristic zero.
- These points are intrinsically defined by the geometry of the curve and do not depend on any embedding or parametrization.
- The sextactic points are invariant under projective transformations that preserve the curve.
Geometric Interpretation
- Sextactic points are the analogues of inflection points, but for conics instead of lines.
- While an inflection point is where the tangent line meets the curve with multiplicity ≥ 3, a sextactic point is where a conic meets the curve with multiplicity ≥ 6 at a single point.
CU Point Validation
- 6P = 2N has 36 solutions,
- Nine of these are flex points, the others 27 are the sextatic points.
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