Ternary Quartics and 3-Dimensional Commutative Algebras

We nd a connection between 3-dimensional commutative algebras with trivial trace and plane quartics and their bitangents. In this paper a structure of a commutative algebra on C 3 is called a 3-dimensional algebra. LetA be the set of 3-dimensional algebras. ConsiderA as a linear space. Let A0 A be the linear subspace of algebras with trivial trace. By denition, 2A0 if the contraction of the structure tensor of is equal to zero. By PV we denote the projectivization of a vector space V. For v2 V; v6 0 we denote by v the corresponding point of the projective space PV . Let 2A0 be an algebra with trivial trace. Recall that an element a2 C 3 is called an idempotent if a6 0; a 2 = a. We say that an element a2 P C 3 = P 2 is a generalized idempotent if a 2 = a , where 2 C. Every idempotent denes a generalized idempotent. Every generalized idempotent a2 P 2 such that a 2 6 0 denes uniquely an idempotent a 0 2 C 3 such that a = a 0 . Dene the subscheme X( ) P 2 of the generalized idempotents by the following equation: