Algebra & Number Theory

Parametrizing quartic algebras over an arbitrary base

Melanie Wood

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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 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 .

Article information

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

Received: 29 June 2010
Revised: 28 September 2010
Accepted: 27 October 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R16: Cubic and quartic extensions
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables

quartic algebras cubic resolvents pairs of ternary quadratic forms degree-4 covers quartic covers


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.

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