Vol. 5, No. 8, 2011

Download this article
Download this article For screen
For printing
Recent Issues

Volume 13
Issue 8, 1765–1981
Issue 7, 1509–1763
Issue 6, 1243–1507
Issue 5, 995–1242
Issue 4, 749–993
Issue 3, 531–747
Issue 2, 251–530
Issue 1, 1–249

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Subscriptions
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
 
Other MSP Journals
Parametrizing quartic algebras over an arbitrary base

Melanie Matchett Wood

Vol. 5 (2011), No. 8, 1069–1094
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 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 .

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