Truncated Grassmannians describe blow-ups of Grassmannians along Schubert varieties, with projection fibers given by spaces of collineations.
Feigin , G _a^M degeneration of flag varieties , Selecta Mathematica 18 :3 (2012), 513--537
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Truncated Grassmannians, blow-ups along Schubert varieties and collineations
Truncated Grassmannians describe blow-ups of Grassmannians along Schubert varieties, with projection fibers given by spaces of collineations.