Orbit Type Strata of C^3: 2-torus Action (a,b)

[HMY]

a 2-torus action on C^3 can be defined by
(a,b).(x,y,z)= (abx, a^-1by, bz)

What are the orbit type strata of C^3 here?

2-torus can be thought of (S^1)^2.
0 is the only fixed point I can tell, so it's one strata.
I just don't understand this seemingly simple action.

 

[matt grime]

Orbits are not fixed points. (0,0,0) is an orbit, agreed. But there is at least one other - every point lies in an orbit.

So fix a point (u,v,w) and look at where the torus maps it. What is the resulting space? It is a quotient space of the 2-torus, but by what? I.e.e when is the map (a,b)-->(abx,by/a,bz) not injective? (this is a constraint on x,y,z) Where it is injective impleis the orbits are 2-toruses, and where it isn't they are something else.

 

[HMY]

am I properly making sense of this?

Call this map f: (a,b)-->(abx,by/a,bz)

f is not injective when you look at (a, b) with b=0 &
a not= 0.

eg.
take another point (c,d) with d=0 & c not= a & c not= 0
So (a,b) not= (c,d). But f(a,b) = (0,0,0) & f(c,d) = (0,0,0)

So fix a point (u,v,w) and look at where the torus maps it. What is the resulting space? It is a quotient space of the 2-torus, but by what? I.e.e when is the map (a,b)-->(abx,by/a,bz) not injective? (this is a constraint on x,y,z) Where it is injective impleis the orbits are 2-toruses, and where it isn't they are something else.

 

[matt grime]

That f is a map from where to where? What is the alleged image? The set of all points (abx,by/a,bz) with x,y,z in C^3?

I don't see what that map has to do with the problem.

Here's a point in C^3: (1,0,0). What is the orbit of that point under group action?