Geodesic
Algebras of invariants of forms that are complete intersections
Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 423-429.

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The author lists all pairs $(n,r)$ such that the algebra of invariants of $n$-forms of degree $r$ is a complete intersection. Under the assumption $n\geqslant2$ and $r\geqslant3$, the pairs are $(2,3)$, $(2,4)$, $(2,5)$, $(2,6)$, $(3,3)$, $(4,3)$, and only these. Bibliography: 12 titles. 
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N. D. Beklemishev. Algebras of invariants of forms that are complete intersections. Izvestiya. Mathematics , Tome 23 (1984) no. 3, pp. 423-429. http://geodesic.mathdoc.fr/item/IM2_1984_23_3_a0/

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