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# On Hilbert Covariants

Published:2012-11-17
Printed: Feb 2014
• Abdelmalek Abdesselam,
Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, VA 22904-4137, USA
• Jaydeep Chipalkatti,
Department of Mathematics, Machray Hall, University of Manitoba, Winnipeg, MB R3T 2N2, Canada
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## Abstract

Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant $\mathcal{H}_{r,d}(F)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $\mathcal{H}$ give defining equations for the image variety $X$ of an embedding $\mathbf{P}^r \hookrightarrow \mathbf{P}^d$. 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 $\mathcal{H}$ 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: binary forms, covariants, $SL_2$-representations
 MSC Classifications: 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 13A50 - Actions of groups on commutative rings; invariant theory [See also 14L24]

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