Polynomial invariants and moduli of generic two-dimensional commutative algebras

Abstract

ABSTRACTLet V be a two-dimensional vector space over a field of characteristic not 2 or 3. We show there is a surjection ν from the set of ‘generic’ commutative algebra structures on V modulo the action of onto . In these coordinates (quotients of invariant quartic polynomials) many properties of the algebra are described by polynomial equations. The map ν is a bijection except over a degenerate elliptic curve Γ, and we give a parametrization of , also characterized as those algebras admitting non-trivial automorphisms, in terms of Galois extensions of . We show ν lifts to a map from the corresponding -orbit space into a hypersurface in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjection from the -orbit space of ‘stable’ commutative algebras onto .