On Hilbert Covariants

Abdesselam, Abdelmalek ; Chipalkatti, Jaydeep

Canadian journal of mathematics, Tome 66 (2014) no. 1, pp. 3-30

Résumé

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.

Détail

DOI : 10.4153/CJM-2012-046-1

Mots-clés : 14L30, 13A50, binary forms, covariants, SL2-representations.