Canonical blow-ups of Grassmannians I: How canonical is a Kausz compactification?

TL;DR

This paper introduces a uniform approach to Kausz compactifications of Grassmannians, revealing their structure as toroidal embeddings and resolving certain birational maps, thus advancing understanding of their geometric properties.

Contribution

It demonstrates that Kausz compactifications are toroidal embeddings and links them to spaces of complete collineations, providing new insights into their structure and resolutions.

Findings

Kausz compactifications are toroidal embeddings of GL(n)

They resolve Landsberg-Manivel birational maps

They connect to spaces of complete collineations

Abstract

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 any 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.