Vol. 5, No. 8, 2011

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ISSN: 1944-7833 (e-only)
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Parametrizing quartic algebras over an arbitrary base

Melanie Matchett Wood

Vol. 5 (2011), No. 8, 1069–1094

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 .

quartic algebras, cubic resolvents, pairs of ternary quadratic forms, degree-4 covers, quartic covers
Mathematical Subject Classification 2000
Primary: 11R16
Secondary: 11E20
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
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
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