Geometric Ramifications of Invariant Expressions in the Binary Hypercommutative Variety

Abstract

The binary hypercommutative (BH) variety, also referred to as the variety of thirdpoint groupoids in Harris (Thirdpoint groupoids, Masters thesis, Virginia State University, 2008), is a collection of algebras associated naturally with the nonsingular points of an irreducible cubic curve. The identities which define these algebras have, as consequences, expressions whose value is invariant under any permutation of the variables comprising the expression. We characterize such expressions and Conjecture that they can be used to determine points of intersection between a given cubic curve and an arbitrary algebraic curve. We prove the Conjecture in the case of the intersection of a cubic γ and a conic β: If γ, β meet in the five points a,b,c,d,e in the real projective plane, then they meet also in the point e*{(a*b)*(c*d)} where a*b denotes the point of intersection besides a,b where the line joining a,b meets γ. The expression e*{(a*b)*(c*d)} is invariant in the BH variety. The Conjecture is also proved for two types of singular cubics. We provide illustrations as strong evidence, but cannot prove similar statements pertaining to the intersection of a nonsingular cubic with higher degree algebraic curves.