Vol. 5, No. 8, 2011

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

Volume 10
Issue 9, 1845–2052
Issue 8, 1601–1843
Issue 7, 1373–1600
Issue 6, 1147–1371
Issue 5, 939–1146
Issue 4, 695–938
Issue 3, 451–694
Issue 2, 215–450
Issue 1, 1–214

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
Cover
Editorial Board
Editors' Addresses
Editors' Interests
About the Journal
Scientific Advantages
Submission Guidelines
Submission Form
Subscriptions
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
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