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On Hilbert Covariants

Published online by Cambridge University Press:  20 November 2018

Abdelmalek Abdesselam  and
Jaydeep Chipalkatti
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.

Keywords

14L3013A50binary formscovariantsSL2-representations.
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 1 , February 2014 , pp. 3 - 30
Copyright
Copyright © Canadian Mathematical Society 2014

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