Let us call an algebraic variety $X$ an *iterated $\mathbb{P}^1$-bundle* if it is either a point or a locally trivial $\mathbb{P}^1$-bundle over $X'$, which is another iterated $\mathbb{P}^1$-bundle.

It is easy to construct examples of such varieties as toric varieties. Another (non-toric in general) examples of such varieties are the Bott-Samelson varieties.

Is there a simple (combinatorial or other) description of these spaces?

I would like this description to allow one to investigate some of the following:

- line bundles (ampleness, global generation etc.),
- properties of the rank two vector bundle $\mathscr{E}$ on $X'$ such that $X = \mathbb{P}(\mathscr{E})$ (which exists and is determined up to multiplication by some line bundle on $X'$),
*sections*of the bundle $X\to X'$ (that is, line subbundles of $\mathscr{E}$),- construction of $\mathbb{P}^1$-bundles over $X$ (i.e., classes of rank two vector bundles),
- degenerations over $\mathbb{A}^1$ to other iterated $\mathbb{P}^1$-bundles (in particular, their toric degenerations).