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The three variables case

Let us first look at the 3-dimensional case. Denote by U the subspace of V spanned by the monomials that do not have z in them, that is

The stabilizer

in G of U induces the full

Proposition. The orbits of

in U are:

Use (cf. [Kraft]): The form represents a cubic curve modulo ``cubes''. First bring the form in Weierstraß normal form and then look at it modulo cubes. A direct proof is given in Appendix 1.

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