Is the Veronese variety "enough" to describe all the $SL(V)$-orbits in $\mathbb{P}(\textrm{Sym}^dV)$?

Question

I apologise in advance if the question will look ridicolous to experienced eyes: in this case a good reference will be enough to clarify my doubts.

Let $V$ be a complex vector space of dimension $n$, and $\textrm{Sym}^dV$ the space of homogeneous polynomial of degree $d$ (on $V^*$). Assume also that $SL_n$ acts naturally on $$\mathbb{P}^{M(n,d)}:=\mathbb{P}(\textrm{Sym}^dV)\, .$$

Motivating the prelim. question. If $n=2$ and $d=3$, then $M(n,d)=3$. In this case, there are two special orbits in $\mathbb{P}^{M(n,d)}=\mathbb{P}^3$: a one-dimensional one, given by the twisted cubic $\gamma$, and a two-dimensional one, given by $T\gamma\smallsetminus\gamma$, which is the set of the smooth points of the tangent variety to $\gamma$. I don't know if $\mathbb{P}^3\smallsetminus T\gamma$ is an orbit as well, but surely it is an invariant subset: let me call it the "open cell".

This example can be easily generalised, by replacing the twisted cubic $\gamma$ with the Veronese variety $v_d(\mathbb{P}(V))$ in $\mathbb{P}^{M(n,d)}$, and by noticing that the subsets $$ X_{i,n,d}:=\textrm{Osc}_i(v_d(\mathbb{P}(V)))\smallsetminus \textrm{Osc}_{i-1}(v_d(\mathbb{P}(V)))\subset \mathbb{P}^{M(n,d)}\quad \quad (^*) $$ are $SL_n$-invariant (by "$\textrm{Osc}_i$" I mean the $i^\textrm{th}$ osculating variety, i.e., the union of the $i^\textrm{th}$ order osculating spaces).

PRELIM. QUESTION: for which values of $n$ and $d$ all the subsets $X_{i,n,d}$ (possibly discarding the "open cell") are orbits?

The example above (if I did it correctly) shows that $n=2$ and $d=3$ are ok, but what next?

Motivating the main question. Observe that $X_{0,n,d}$ is the Veronese itself, which is always an orbit (as long as $SL_n$ acts transitively on $\mathbb{P}V$). Observe also that the next $X_{i,n,d}$'s are constructed from $X_{0,n,d}$ in a "natural way" (e.g., by taking tangent lines).

Idea. Maybe there are more sophisticated "natural operations" allowing to "enlarge" the Veronese variety $X_{0,n,d}$ in such a way that the difference between subsequent "enlargements" is an orbit.

I call "natural enlargement" the construction of a larger invariant subset out of a smaller one, by means of natural objects associated to the latter (lines/subspaces with a certain tangency, (multi)secant subspaces, sections of natural bundles, etc.).

MAIN QUESTION: Is there any other example of a decomposition of $\mathbb{P}^{M(n,d)}$ into invariant subsets, simliar to $(^*)$, but finer?

By "similar" I mean a formula like $(^*)$ where the osculating varieties are replaced by something else, and by "finer" I mean that the so-obtained invariant subsets are smaller than the $X_{i,n,d}$'s.

Answer 1

one can take the secant varieties of the Veronse, where in general $\sigma_r(X)=\overline{\langle x_1,...,x_r\rangle \mid x_j\in X}$, where $\langle \rangle$ denotes linear span. These sets are $GL(V)$-invariant, as are the sets constructed by mixing your osculating construction and these (e.g. take $X$ the tangential variety of the Veronese instead of just the Veronese). The cases where they are all orbits are just $d=2$ any $n$ or $d=3$ $n=2$. Another way to get invariant sets is to take the dual variety of the Veronese, then its singular locus, then the singular locus of the singular locus etc.. and again one can mix and match constructions.