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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1$\begingroup$ I think the notion you are looking for is the join of two varieties. This is discussed in Harris' First Course. $\endgroup$– Tabes BridgesApr 12, 2016 at 16:18
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$\begingroup$ Another generalization is a projectivization of a vector bundle. $\endgroup$– SashaApr 12, 2016 at 16:27
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$\begingroup$ @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 $Y$. $\endgroup$– user46071Apr 12, 2016 at 16:42
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