## Algebra & Number Theory

### Parametrizing quartic algebras over an arbitrary base

Melanie Wood

#### Abstract

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree-4 $S$-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 $ℤ$ 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 $ℤ$.

#### Article information

Source
Algebra Number Theory, Volume 5, Number 8 (2011), 1069-1094.

Dates
Revised: 28 September 2010
Accepted: 27 October 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729734

Digital Object Identifier
doi:10.2140/ant.2011.5.1069

Mathematical Reviews number (MathSciNet)
MR2948473

Zentralblatt MATH identifier
1271.11043

#### Citation

Wood, Melanie. Parametrizing quartic algebras over an arbitrary base. Algebra Number Theory 5 (2011), no. 8, 1069--1094. doi:10.2140/ant.2011.5.1069. https://projecteuclid.org/euclid.ant/1513729734

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