Parametrizing quartic algebras over an arbitrary base

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 $\mathbb{Z}$.