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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).
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Hi Piotr -- you might add "effectiveness" of line bundles to your first item... though I expect such a thing might be complicated in general. – Dave Anderson Oct 22 '10 at 15:20
You cannot hope to have a simple combinatorial description --- the theory of rank 2 vector bundles on $P^1\times P^1$ is already rather complicated. – Sasha Oct 22 '10 at 18:53
Dear Sasha, that's a good point! But maybe, as in the framework of toric varieties, we can restrict ourselves to some "uniform" or "equivariant" rank two bundles, in which case there would be some good description (a class wide enough to include examples that appear in nature)? Note that every uniform rank two vector bundle on the quadric splits and that equivariant bundles on toric varieties have a good description. – Piotr Achinger Oct 22 '10 at 21:56
Certainly, if you restrict to the toric case, a combinatorial description is possible and I guess is quite simple. – Sasha Oct 23 '10 at 3:12
But the point is that e.g. the Bott-Samelson varieties are not toric in general, and I would like to include them as well. Probably I should consider such towers with a $B$-action instead of $T$-action (the toric ones would then have this $B$-action factored through the projection $B\to T$). – Piotr Achinger Oct 23 '10 at 13:33

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.)

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Actually, Pasquier's paper is a partial motivation for this question. His degeneration however relies heavily of the BSDH varieties being a quotient by an action of a product of Borels (he ,,degenerates'' the identity $B\to B$ into the projection $B\to T$). – Piotr Achinger Oct 22 '10 at 17:33

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