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- W2950954994 abstract "Let $V$ be a two-dimensional vector space over a field $mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $nu$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of $GL(V)$ onto the plane $mathbb F^2$. In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map $nu$ is a bijection over the complement of a degenerate elliptic curve $Gamma$ and over $Gamma$ we give an explicit parametrisation of the fibre in terms of Galois extensions of $mathbb F$. Algebras in $nu^{-1}(Gamma)$ are exactly those which admit non-trivial automorphisms. We show how $nu$ can be lifted to a map from the $SL(V)-$moduli space to an algebraic hypersurface $Gamma'$ 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 surjective map from the set of stable commutative algebras on $V$ modulo the action of $SL(V)$ onto $Gamma'$." @default.
- W2950954994 created "2019-06-27" @default.
- W2950954994 creator A5025800314 @default.
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- W2950954994 date "2016-12-17" @default.
- W2950954994 modified "2023-09-25" @default.
- W2950954994 title "Polynomial invariants and moduli of generic two-dimensional commutative algebras" @default.
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- W2950954994 doi "https://doi.org/10.48550/arxiv.1612.05737" @default.
- W2950954994 hasPublicationYear "2016" @default.
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