Parametrizing quartic algebras over an arbitrary base

Wood, Melanie

doi:10.2140/ant.2011.5.1069

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Parametrizing quartic algebras over an arbitrary base

Melanie Matchett Wood

Vol. 5 (2011), No. 8, 1069–1094

DOI: 10.2140/ant.2011.5.1069

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 $ℤ$ .

Keywords

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

Mathematical Subject Classification 2000

Primary: 11R16

Secondary: 11E20

Milestones

Received: 29 June 2010

Revised: 28 September 2010

Accepted: 27 October 2010

Published: 5 June 2012

Proposed: Hendrik W. Lenstra

Seconded: Raman Parimala, Bernd Sturmfels

Authors

Melanie Matchett Wood
Department of Mathematics University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53705 United States	American Institute of Mathematics 360 Portage Ave Palo Alto, CA 94306 United States

Vol. 5, No. 8, 2011