Parametrizing quartic algebras over an arbitrary base

TL;DR

This paper generalizes the parametrization of quartic algebras with cubic resolvents from integers to arbitrary base schemes using pairs of ternary quadratic forms, providing a geometric construction that is base change compatible.

Contribution

It extends Bhargava's classical parametrization to arbitrary bases and offers a geometric construction that aligns with existing explicit methods.

Findings

Provides a universal parametrization over any base scheme.

Constructs quartic algebras from pairs of ternary quadratic forms.

Ensures the construction commutes with base change.

Abstract

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 $\mathbb{Z}$.