Truncated Grassmannians, blow-ups along Schubert varieties and collineations

TL;DR

This paper explores the geometric structures called truncated Grassmannians, their relation to blow-ups of flag varieties along Schubert varieties, and describes fibers via collineation spaces.

Contribution

It introduces a new framework linking truncated Grassmannians to blow-ups of flag varieties and details the fiber structure in the Grassmannian case.

Findings

Truncated Grassmannians relate to blow-ups along Schubert subvarieties.

Blow-ups form a larger family projecting onto Grassmannians.

Fibers are described via spaces of collineations.

Abstract

Truncated Grassmannians are defined as closures of orbits of abelian unipotent groups acting on the degree truncations of projectivized wedge powers. We show that such truncations in a more general setup show up in the description of the blow-ups of general flag varieties along Schubert subvarieties. We work out the case of Grassmannians in detail. In particular, we show that our blow-ups are members of a larger family of varieties projecting onto Grassmannians, and describe the fibers of these projections via the spaces of collineations.