If I understand correctly, in geometric invariant theory, polystable points can be defined as those which have a closed orbit. Is it true that semistable points can be characterized as those whose orbit closure contains a unique polystable orbit?


Edit: It would seem that polystable points are actually those whose orbit are closed *in the semistable locus*. So my pseudo-definition of semistability is circular, unless an independent definition of polystability is given. Also, I suppose that since the unstable (non semistable) locus is closed, there is no way the closure of an unstable orbit could intersect a polystable (or semistable) orbit. I guess that answers my question. But if someone has more insights, please feel free to write them.



Semistability in GIT
If I may answer my own question:


1. Polystable points can be characterized as points that are semistable and with closed orbit in the semistable locus.
2. Yes, If the orbit closure of a point contains a polystable orbit (necessarily unique), then that point is semistable.


However 2. is pretty much saying nothing. In particular unstable orbits can be closed.


