TITLE: Structure of iterated $\mathbb{P}^1$-bundles
QUESTION [5 upvotes]: 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).

REPLY [2 votes]: This is only a partial answer, but the paper
M. Willems, "K-théorie équivariante des tours de Bott", Duke Math. J. 132 (2006)
includes a combinatorial description of a degeneration of Bott-Samelson varieties to toric ones.  (It's based on a construction of Grossberg and Karshon, and was also done algebraically by Pasquier.)