An Invariant of Algebraic Curves from the Pascal Theorem

TL;DR

This paper introduces a new geometric invariant of algebraic curves inspired by Pascal's theorem, enabling a generalization of Pascal's theorem to cubic and higher degree curves with simpler conditions than classical theorems.

Contribution

A novel geometric invariant is discovered that generalizes Pascal's theorem to cubic and higher degree algebraic curves, simplifying classical geometric theorems.

Findings

Generalization of Pascal's theorem to cubic curves.

The new invariant simplifies classical theorems like Chasles's and Cayley-Bacharach.

The generalized theorem involves nine intersection points and a Pascal mapping.

Abstract

In 1640's, Blaise Pascal discovered a remarkable property of a hexagon inscribed in a conic - Pascal Theorem, which gave birth of the projective geometry. In this paper, a new geometric invariant of algebraic curves is discovered by a different comprehension to Pascal's mystic hexagram or to the Pascal theorem. Using this invariant, the Pascal theorem can be generalized to the case of cubic (even to algebraic curves of higher degree), that is, {\em For any given 9 intersections between a cubic $\Gamma_3$ and any three lines $a,b,c$ with no common zero, none of them is a component of $\Gamma_3$, then the six points consisting of the three points determined by the Pascal mapping applied to any six points (no three points of which are collinear) among those 9 intersections as well as the remaining three points of those 9 intersections must lie on a conic.} This generalization differs quite a bit and is much simpler than Chasles's theorem and Cayley-Bacharach theorems.