Algebra & Number Theory

Parametrizing quartic algebras over an arbitrary base

Melanie Wood

Full-text: Open access

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 .

Article information

Source
Algebra Number Theory, Volume 5, Number 8 (2011), 1069-1094.

Dates
Received: 29 June 2010
Revised: 28 September 2010
Accepted: 27 October 2010
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729734

Digital Object Identifier
doi:10.2140/ant.2011.5.1069

Mathematical Reviews number (MathSciNet)
MR2948473

Zentralblatt MATH identifier
1271.11043

Subjects
Primary: 11R16: Cubic and quartic extensions
Secondary: 11E20: General ternary and quaternary quadratic forms; forms of more than two variables

Keywords
quartic algebras cubic resolvents pairs of ternary quadratic forms degree-4 covers quartic covers

Citation

Wood, Melanie. Parametrizing quartic algebras over an arbitrary base. Algebra Number Theory 5 (2011), no. 8, 1069--1094. doi:10.2140/ant.2011.5.1069. https://projecteuclid.org/euclid.ant/1513729734


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References

  • M. Bhargava, “Higher composition laws, II: On cubic analogues of Gauss composition”, Ann. of Math. $(2)$ 159:2 (2004), 865–886.
  • M. Bhargava, “Higher composition laws, III: The parametrization of quartic rings”, Ann. of Math. $(2)$ 159:3 (2004), 1329–1360.
  • M. Bhargava, “The density of discriminants of quartic rings and fields”, Ann. of Math. $(2)$ 162:2 (2005), 1031–1063.
  • G. Casnati, “Covers of algebraic varieties. III. The discriminant of a cover of degree $4$ and the trigonal construction”, Trans. Amer. Math. Soc. 350:4 (1998), 1359–1378.
  • G. Casnati and T. Ekedahl, “Covers of algebraic varieties. I. A general structure theorem, covers of degree $3,4$ and Enriques surfaces”, J. Algebraic Geom. 5:3 (1996), 439–460.
  • H. Davenport and H. Heilbronn, “On the density of discriminants of cubic fields. II”, Proc. Roy. Soc. London Ser. A 322:1551 (1971), 405–420.
  • P. Deligne, letter to W. T. Gan, B. Gross and G. Savin, November 13 2000.
  • P. Deligne, letter to M. Bhargava, March 5 2004.
  • B. N. Delone and D. K. Faddeev, \cyr Teoriya irracional'nosteĭ tret'eĭ stepeni, Trudy Mat. Inst. Steklov 11, 1940. Translated as The theory of irrationalities of the third degree, Trans. Math. Monographs 10, Amer. Math. Soc., Providence, 1964. Original at http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=900.
  • A. Grothendieck, “Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, II”, Inst. Hautes Études Sci. Publ. Math. 17 (1963), 137–223.
  • D. Eisenbud, The geometry of syzygies: a second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics 229, Springer, New York, 2005.
  • W. T. Gan, B. Gross, and G. Savin, “Fourier coefficients of modular forms on $G\sb 2$”, Duke Math. J. 115:1 (2002), 105–169.
  • D. W. Hahn and R. Miranda, “Quadruple covers of algebraic varieties”, J. Algebraic Geom. 8:1 (1999), 1–30.
  • R. Hartshorne, Residues and duality: lecture notes of a seminar on the work of A. Grothendieck, Lecture Notes in Math. 20, Springer, Berlin, 1966.
  • R. Miranda, “Triple covers in algebraic geometry”, Amer. J. Math. 107:5 (1985), 1123–1158.
  • B. Poonen, “The moduli space of commutative algebras of finite rank”, J. Eur. Math. Soc. $($JEMS$)$ 10:3 (2008), 817–836.
  • J. Voight, “Rings of low rank with a standard involution”, preprint, 2010. to appear in Ill. J. Math.
  • C. A. Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, Cambridge University Press, 1994.
  • M. M. Wood, “Gauss composition over an arbitrary base”, Adv. Math. 226:2 (2011), 1756–1771.
  • M. M. Wood, “Rings and ideals parameterized by binary $n$-ic forms”, J. Lond. Math. Soc. $(2)$ 83:1 (2011), 208–231.