On the Jacobian ideal of the binary discriminant

• Autores: Carlos D'Andrea , Jaydeep V. Chipalkatti
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 58, Fasc. 2, 2007, págs. 155-180
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • Let $\Delta$ denote the discriminant of the generic binary $d$-ic. We show that for $d \geq 3$, the Jacobian ideal of $\Delta$ is perfect of height $2$. Moreover we describe its $SL_2$-equivariant minimal resolution and the associated differential equations satisfied by $\Delta$. A similar result is proved for the resultant of two forms of orders $d,e$ whenever $d \geq e - 1$. If $\Phi_n$ denotes the locus of binary forms with total root multiplicity $\geq d - n$, then we show that the ideal of $\Phi_n$ is also perfect, and we construct a covariant which characterizes this locus. We also explain the role of the Morley form in the determinantal formula for the resultant. This relies upon a calculation which is done in the appendix by A. Abdesselam.

• Referencias bibliográficas
  • A. Abdesselam and J. Chipalkatti, Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture (math.AG/0411110), to appear...
  • C. D’Andrea and A. Dickenstein, Explicit formulas for the multivariate resultant, J. Pure Appl. Algebra 164 (2001), 59–86.
  • E. Arbarello, M. Cornalba, P.A. Griffiths, and J. Harris, Geometry of Algebraic Curves, I, Grundlehren der Mathematischen Wissenschaften 267,...
  • M.F. Atiyah and I.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Publishing Co., Reading, Mass.-London-Don Mills, Ont....
  • R. Bott and L.W. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin 1982.
  • W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Mathematics 1327, Springer-Verlag, Berlin, 1988.
  • J.V. Chipalkatti, On equations defining coincident root loci, J. Algebra 267 (2003), 246–271.
  • J.A. Dieudonné and J.B. Carrell, Invariant Theory, Old and New, Academic Press, New York-London, 1971.
  • I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Notes Series 296, Cambridge University Press, Cambridge, 2003.
  • D. Eisenbud, Green’s conjecture: an orientation for algebraists, in Free resolutions in commutative algebra and algebraic geometry (Sundance,...
  • W. Fulton and J. Harris, Representation Theory, A First Course, Graduate Texts in Mathematics, Springer-Verlag, 1991.
  • I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston, Inc.,...
  • O. Glenn, The Theory of Invariants, Ginn and Co., Boston, 1915, (Available as an eBook from “Project Gutenberg” at [www.gutenberg.net](http://www.gutenberg.net).)
  • J.H. Grace and A. Young, The Algebra of Invariants, 1903, Reprinted by Chelsea Publishing Co., New York, 1962.
  • R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1977.
  • D. Hilbert, Ueber die nothwendigen und hinreichenden covarianten Bedingungen für die Darstellbarkeit einer binären Form als vollständiger...
  • D. Hilbert, Ueber die Singularitäten der Discriminantenfläche, Math. Ann. 30 (1887), 437–441.
  • J.P. Jouanolou, Formes d’inertie et résultant: un formulaire, Adv. Math. 126 (1997), 119–250.
  • J. Kung and G.-C. Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. N.S. 10 (1984), 27–85.
  • P. Olver, Classical Invariant Theory, London Mathematical Society Student Texts, Cambridge University Press, Cambridge, 1999.
  • G. Salmon, Higher Algebra, Reprinted by Chelsea Publishing Co., New York, 1964.
  • B. Sturmfels, Algorithms in Invariant Theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993.