Varieties swept out by grassmannians of lines

TL;DR

This paper classifies high-dimensional complex projective varieties that are covered by families of Grassmannians of lines, identifying their structure as fibrations or specific Grassmannian varieties.

Contribution

It provides a classification of varieties swept out by Grassmannians of lines, extending to cases with linear spaces and quadrics for certain dimensions.

Findings

Varieties are either fibrations over surfaces with Grassmannian fibers or are Grassmannians themselves.

Classifies varieties for dimensions $2r \,\geq\, 8$.

Includes special cases for $r=2$ and $r=3$ with linear spaces and quadrics.

Abstract

We classify complex projective varieties of dimension $2r \geq 8$ swept out by a family of codimension two grassmannians of lines $\mathbb{G}(1,r)$. They are either fibrations onto normal surfaces such that the general fibers are isomorphic to $\G(1,r)$ or the grassmannian $\mathbb{G}(1,r+1)$. The cases $r=2$ and $r=3$ are also considered in the more general context of varieties swept out by codimension two linear spaces or quadrics.