Notes on Filip’s Proof That Orbit Closures Are Algebraic

One should expect any loci cut out by such algebro-geometric conditions to be a variety by basic reasons, and indeed to prove orbit closures are varieties Filip explains that it suffices to show that they are cut out by such conditions. (Filip expands the list slightly.) By two deep theorems of Möller, every closed orbit is known to be described by such conditions, which you might think would give hope that verifying a similar result for all orbit closures might be possible. But on the other hand, Möller’s proof concerning torsion, although short, is only accessible to those with a truly formidable background in algebraic geometry, and it is not even clear what the statement of a generalization of this result to all orbit closures would look like. And perhaps most importantly, Möller’s work in fact used that closed orbits are varieties! (Smillie’s Theorem gives that closed orbits are line bundles over algebraic curves.)