Consider the linear representation $V=$ Sym$^3(\mathbb{C}^3)$ of SL$_3(\mathbb{C})$. Now Theorem 7.2 of Popov-Vinberg asserts the following (Richardson [1972a], Luna [1973]): For any action of a reductive (linear) algebraic group $G$ on an irreducible smooth affine variety X there exists the stabiliser in general position. Now SL$_3(\mathbb{C})$ is known to be reductive and furthermore $V$ is irreducible and it can be identified with $\mathbb{C}[X,Y,Z]_3$ the cubic ternary forms. So I have existence of the stabiliser in general position.
My question is this: this example should be easy enough to show this explicitly and actually calculate what the generic stabiliser is. How does one go about finding a Zariski dense subset for which all the stabilisers are conjugate in this case?
Now I have a few thoughts on this: by a character argument, the stabiliser of the linear action should coincide with that of the corresponding action of $G$ on $\mathbb{P}(V)$ on the homogeneous forms corresponding to plane curves in $\mathbb{P}^2$ (this may not be needed). Furthermore the only stable plane curves correspond with the smooth locus of elliptic curves (see for example Dolgachev's book on invariant theory Chapter 10), so logically the smooth locus should be a good candidate for the Zariski open and the s.g.p. (stabiliser in general position) should also be a finite group.
One would get the result above if one could compute all the stabilisers of Elliptic curves in Weierstrass normal form say, but then one would still need to show these are all conjugate trying to use this linear section. By hand computation of any of these seems hard however.
Perhaps there is a reduction technique that can be used, perhaps looking at the stabiliser of the standard Borel or perhaps passing to the Lie algebra action although I have not got these arguments to work... It also appears one can reduce to the action of the Weyl group on a linear section for the computation of the actual algebra of invariants (Popov-Vinberg page 133).
If one checks the stabiliser $G_f$ where $f: x^3+y^3+z^3$ has non-vanishing discriminant and so is smooth then one would assume that the subgroup of SL$_3(\mathbb{C})$ which stabilises this would include the permutations of the $x,y,z$ and the scaling of each by a cube root of unity $\zeta$ say... so $S_3 \times C_3^3$ at least.
Any suggestions on how one should approach such a problem would be most appreciated.