As far as I can see, scrolls are constructed by means of rational curves. Is it possible a generalization with algebraic varieties of arbitrary dimension?

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    $\begingroup$ I think the notion you are looking for is the join of two varieties. This is discussed in Harris' First Course. $\endgroup$ – Tabes Bridges Apr 12 '16 at 16:18
  • $\begingroup$ Another generalization is a projectivization of a vector bundle. $\endgroup$ – Sasha Apr 12 '16 at 16:27
  • $\begingroup$ @TabesBridges: the join of two varieties $X$ and $Y$ consists of all the lines joining $X$ and $Y$, I want only the lines of type $P,P^{\phi}$, with $\phi$ an isomorphism between $X$ and $Y$. $\endgroup$ – user46071 Apr 12 '16 at 16:42

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