Parametrizing quartic algebras over an arbitrary base

2010/07/30 by Melanie Matchett Wood, Wood, Melanie Matchett · 2 citations
Mathematics · #11E20 #11R16 #14D23 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11E20 #msc:11R16 #msc:14D23

paper · pdf · doi:10.48550/arxiv.1007.5503

submitted

arxiv created 2010/07/30 · arxiv updated 2010/08/30

Abstract

We parametrize quartic commutative algebras over any base ring or scheme (equivalently finite, flat degree four 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.

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