Parametrizing quartic algebras over an arbitrary base

Details Parametrizing quartic algebras over an arbitrary base Parametrizing quartic algebras over an arbitrary base

2010-07-30

arxiv.org/abs/1007.5503v1

Mathematics

Number Theory

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Scientific paper

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four $S$-schemes), with their cubic resolvents, by pairs of ternary quadratic forms over the base. This generalizes Bhargava's parametrization of quartic rings with their cubic resolvent rings over $\mathbb{Z}$ by pairs of integral ternary quadratic forms, as well as Casnati and Ekedahl's construction of Gorenstein quartic covers by certain rank 2 families of ternary quadratic forms. We give a geometric construction of a quartic algebra from any pair of ternary quadratic forms, and prove this construction commutes with base change and also agrees with Bhargava's explicit construction over $\mathbb{Z}$.

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