Orbits of Algebraic Group Actions Are Locally Closed
Let a linear algebraic group \(G\) act regularly on an algebraic variety \(X\). For every \(x \in X\), the orbit \(Gx\) is a locally closed subvariety of \(X\), open in its closure. The boundary \(\overline{Gx}\setminus Gx\), if nonempty, is a union of orbits of strictly smaller dimension. In particular, every orbit closure contains at least one closed orbit.
Proof
The orbit map \(a_x:G\to X\), \(g\mapsto gx\), has constructible image by Chevalley's theorem. Let \(Y=\overline{Gx}\). A constructible dense subset of an irreducible component of \(Y\) contains a nonempty open subset \(U\). Translates of \(U\) by elements of \(G\) remain in the same orbit and cover \(Gx\), so \(Gx\) is open in \(Y\).
Since the orbit is open in its closure, any orbit contained in \(Y\setminus Gx\) has smaller dimension. Repeating the dimension drop in a Noetherian space reaches a closed orbit.