Let $k$ be a field, and let $T$ an $n$-dimensional split torus over $k$. Let $X$ be a $k$-scheme with algebraic $T$-action. Solve for X:

$$X / T \cong \mathbf{P}^1_k$$

(The quotient should be a categorical quotient in the category of $k$-schemes. A categorical quotient is a morphism $\pi : X \to X/T$ which is universal among morphisms which are constant on $T$-orbits.)

For example, $X = \mathbf{P}^1_k \times T$ with $T$ acting by right multiplication on the right factor is a trivial solution. More generally, I expect to find such $X$ among $(n+1)$-dimensional toric varieties.

Would someone explain how nasty this question actually is and what restrictions could be made:

- on the field $k$ or
- on the scheme $X$ or
- on the quotient morphism $\pi : X \to X/T$ or
- on the action morphism $\gamma : X \times T \to X$ or
- on the category in which the quotient is taken or
- on the "goodness" of the categorical quotient

in order to have a simple classification of the solutions? In other words, the solution to this problem will be itself a problem, a possibly modified version of the above admitting a simple solution. This might be a complex question (or not!), in which case a partial answer would still be interesting.