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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.

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    $\begingroup$ If $\dim V = 2$ and $d=4$, then $\mathrm{Sym}^d(V)$ is degree $4$ homegenous polynomials in two variables. Such a polynomial has $4$ roots in $\mathbb{P}^1$. Let $\lambda$ be the cross ratio of these roots (this depends on choosing an order of the roots). Then the $j$-invariant, $\tfrac{256 (1-\lambda+\lambda^2)^3}{\lambda^2 (1-\lambda)^2}$ is independent of the order of the roots and is invariant for the $SL(V)$ action. This gives uncounteably many orbits, if your ground field is uncountable, so I am pessimistic about a simple description of the orbits in general. $\endgroup$ Nov 6, 2015 at 15:00
  • $\begingroup$ @DavidSpeyer Nice remark! However, me neither ever believed in this 'simple description' of the orbits: it is just an idealistic goal I used to clarify what I had in mind. What I'm really interested about is to get to know methods for producing invariant subsets out of the Veronese variety, which 'contain as few orbits as possible'. For instance, these infinite orbits of yours, where do they sit? Certainly outside the Veronese $v_4(\mathbb{P}^1)$, which is made of polynomials with quadruple roots. But how they interact with the tangent/osculating variety? Are they inside/outside/between them?? $\endgroup$ Nov 6, 2015 at 17:16

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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.

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