$\mathbb Z$ acts on the lattice $\mathbb Z \times \mathbb Z$ by adding an element to itself n times. 


I am studying some function arising from symplectic geometry which happens in my case to be naturally defined on $\mathbb Z \times \mathbb Z$ and is constant on each orbit of the $\mathbb Z$-action, thus it lives ob the orbit space.


I am curious what kinds of algebraic/geometric/arithmetic structures does this set or orbits has, to compare with the behaviour of my function.


Thanks



Does $\mathbb Z \times \mathbb Z$ mod the obvious $\mathbb Z$ action have more structure than just a set?
The set $(\mathbb Z\times \mathbb Z)\setminus \{(0,0)\}$ modulo the equivalence relation $(x,y)\sim(nx,ny)$, for $n\in \mathbb Z\_{\not =0}$, is known as $\mathbb Q\mathbb P^1$, the projective line over $\mathbb Q$. It's an algebraic variety over the rationals (or rather, it's more correct to say that it's the $\mathbb Q$-points of an algebraic variety over the rationals).


The set $(\mathbb Z\times \mathbb Z)\setminus \{(0,0)\}$ modulo $(x,y)\sim(nx,ny)$, for $n\in \mathbb Z\_{> 0}$, is not the $\mathbb Q$-points of an algebraic variety in any obvious way.


