Parametrizing quartic algebras over an arbitrary base

@article{Wood2010ParametrizingQA,
title={Parametrizing quartic algebras over an arbitrary base},
author={M. Wood},
journal={arXiv: Number Theory},
year={2010}
}

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… CONTINUE READING