#### Vol. 5, No. 8, 2011

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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 $ℤ$ 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
Primary: 11R16
Secondary: 11E20