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In this paper, we develop a simple uniform picture incorporating the Kausz compactifications and the spaces of complete collineations by blowing up Grassmannians   $G(p,n)$   according to a torus action   $\mathbb G_{m}$  . We show that each space of complete collineations is isomorphic to a maximal-dimensional connected component of the   $\mathbb G_{m}$  -fixed point scheme of a Kausz-type compactification. We prove that the Kausz-type compactification is the total family over the Hilbert quotient   $G(p,n)/ \! \! / \mathbb G_{m}$  , which is isomorphic to the space of complete collineations. In particular, the Kausz compactifications are toroidal embeddings of general linear groups in the sense of Brion–Kumar. We also show that the Kausz-type compactifications resolve the Landsberg–Manivel birational maps from projective spaces to Grassmannians, by comparing Kausz’s construction with ours. As an application, by studying the foliation we derive resolutions of certain birational maps among projective bundles over Grassmannians. The results in this paper are partially taken from the first author’s earlier arxiv post (Canonical blow-ups of Grassmann manifolds, arxiv:2007.06200), which has been revised and expanded in collaboration with the second author.

Canonical Blow-Ups of Grassmannians I: How Canonical Is a Kausz Compactification?