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On Hilbert Covariants

Published online by Cambridge University Press:  20 November 2018

Abdelmalek Abdesselam
Affiliation:
Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904-4137, USA e-mail: [email protected]
Jaydeep Chipalkatti
Affiliation:
Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904-4137, USA e-mail: [email protected]
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Abstract

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Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant ${{H}_{r,\,d}}\,\left( F \right)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $H$ give defining equations for the image variety $X$ of an embedding ${{\text{P}}^{r}}\,\to \,{{\text{P}}^{d}}$. In this paper we describe a new construction of the Hilbert covariant and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $H$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2014

References

[1] Abdesselam, A., On the volume conjecture for classical spin networks. J. Knot Theory Ramifications 21, 1250022, 2012. http://dx.doi.org/10.1142/S0218216511009522 Google Scholar
[2] A. Abdesselam, and Chipalkatti, J., Brill–Gordan loci, transvectants and an analogue of the Foulkes conjecture. Adv. Math. 208(2007), 491520. http://dx.doi.org/10.1016/j.aim.2006.03.003 Google Scholar
[3] A. Abdesselam, , On the Wronskian combinants of binary forms. J. Pure Appl. Algebra 210(2007), 4361. http://dx.doi.org/10.1016/j.jpaa.2006.08.006 Google Scholar
[4] Aluffi, P. and Faber, C., Linear orbits of d-tuples of points in P1. J. Reine Angew. Math. 445(1993), 205220.Google Scholar
[5] Arbarello, E., Cornalba, M., Griffiths, P.A., and Harris, J., Geometry of Algebraic Curves, Vol. I. Grundlehren Math.Wiss. 267, Springer-Verlag, New York, 1985.Google Scholar
[6] Briand, E., Covariants vanishing on totally decomposable forms. In: Liaison, Schottky Problem and Invariant Theory, Progr. Math. 280, Birkh–user, Basel, 2010, 237256.Google Scholar
[7] Brioschi, F., Sopra un teorema del sig. Hilbert. Rendiconti del Circolo Matematico di Palermo tomo X(1896), 153157.Google Scholar
[8] Chipalkatti, J., On Hermite's invariant for binary quintics. J. Algebra 317(2007), 324353. http://dx.doi.org/10.1016/j.jalgebra.2007.06.021 Google Scholar
[9] Chipalkatti, J., On the saturation sequence of the rational normal curve. J. Pure Appl. Algebra 214(2010), 15981611. http://dx.doi.org/10.1016/j.jpaa.2009.12.005 Google Scholar
[10] Clebsch, A., Ueber symbolische Darstellung algebraischer Formen. J. Reine Angew. Math. vol. 59, pp. 1–62, 1861.Google Scholar
[11] Dolgachev, I., Lectures on Invariant Theory. London Math. Soc. Lecture Note Ser. 296, Cambridge University Press, 2003.Google Scholar
[12] Dolgachev, I., Classical Algebraic Geometry: A Modern View. Cambridge University Press, Cambridge, 2012.Google Scholar
[13] Eisenbud, D., Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Math. 150, Springer-Verlag, New York, 1995.Google Scholar
[14] Eisenbud, D., The Geometry of Syzygies. Graduate Texts in Math. 229, Springer-Verlag, New York, 2005.Google Scholar
[15] Fisher, C. S., The death of a mathematical theory: A study in the sociology of knowledge. Arch. Hist. Exact Sci. 3(1966), 137159. http://dx.doi.org/10.1007/BF00357267 Google Scholar
[16] Fulton, W. and Harris, J., Representation Theory, A First Course. Graduate Texts in Math. 129 Springer-Verlag, New York, 1991.Google Scholar
[17] Glenn, O., The Theory of Invariants. Ginn and Co., Boston, 1915.Google Scholar
[18] Grace, J. H. and Young, A., The Algebra of Invariants. Chelsea Publishing Co., New York, 1962.Google Scholar
[19] Hilbert, D.,Ueber die notwendigen und hinreichenden covarianten Bedingungen für die Darstellbarkeit einer binären Form als vollständiger Potenz. Math. Ann. 27(1886), 158161. http://dx.doi.org/10.1007/BF01447309 Google Scholar
[20] Hilbert, D., Theory of Algebraic Invariants. Cambridge University Press, Cambridge, 1993.Google Scholar
[21] Knapp, A., Lie Groups Beyond an Introduction. Second edition. Birkhäuser, Boston, 2002.Google Scholar
[22] Kung, J. P. S. and Rota, G.-C., The invariant theory of binary forms. Bull. Amer. Math. Soc. (N.S.) 10(1984), 2785. http://dx.doi.org/10.1090/S0273-0979-1984-15188-7 Google Scholar
[23] Littlewood, D. E., Invariant theory, tensors and group characters. Philos. Trans. Roy. Soc. London Ser. A 239(1944), 305365. ttp://dx.doi.org/10.1098/rsta.1944.0001 Google Scholar
[24] MacDonald, I. G., Symmetric Functions and Hall polynomials. Second edition. Oxford University Press, New York, 1995.Google Scholar
[25] Meulien, M., Sur la complication des algèbres d‘invariants combinants. J. Algebra 284(2005), 284295. http://dx.doi.org/10.1016/j.jalgebra.2004.08.003 Google Scholar
[26] Olver, P., Classical Invariant Theory. London Math. Soc. Stud. Texts, Cambridge University Press, Cambridge, 1999.Google Scholar
[27] Processi, C., Lie Groups, an Approach Through Invariants and Representations. Universitext, Springer-Verlag, New York, 2007.Google Scholar
[28] Rota, G.-C., Two Turning Points in Invariant Theory. Math. Intelligencer 21(1999), 2027. http://dx.doi.org/10.1007/BF03024826 Google Scholar
[29] Salmon, G., Exercises in the hyperdeterminant calculus. Cambridge and Dublin Math. J. 9(1854), 1933.Google Scholar
[30] Salmon, G., Lessons Introductory to Higher Algebra. Chelsea Publishing Co., New York, 1965.Google Scholar
[31] B. Sturmfels, , Algorithms in Invariant Theory. Texts Monogr. Symbol. Comput., Springer-Verlag, Wien–New York, 1993.Google Scholar