The automorphism groups of generalized Kausz compactifications and spaces of complete collineations

TL;DR

This work classifies automorphism groups for generalized Kausz compactifications $\mathcal T_{s,p,n}$ and generalized spaces of complete collineations $\mathcal M_{s,p,n}$. The authors combine Mille Cr\'epe coordinates with Brion-type results on effective cones and the positivity properties of the anticanonical bundle to reduce the problem to the action on ambient Grassmannians and related projective bundles, then identify explicit group structures and exceptional symmetries. They prove that $-K_{\mathcal M_{s,p,n}}$ is ample, while $-K_{\mathcal T_{s,p,n}}$ is big and nef (ample only when the rank $r\le 2$), and provide complete descriptions of ${\rm Aut}(\mathcal M_{s,p,n})$ and ${\rm Aut}(\mathcal T_{s,p,n})$ with detailed case analyses involving USD, DUAL, and Usd operations. These results extend the automorphism classifications for related moduli spaces and connect to the broader theory of wonderful and spherical varieties in arithmetic and representation-theoretic contexts.

Abstract

In this paper, we determine the automorphism groups of generalized Kausz compactifications $\mathcal T_{s,p,n}$. By establishing the (semi-)positivity of the anticanonical bundles of $\mathcal T_{s,p,n}$, we also determine the automorphism groups of generalized spaces of complete collineations $\mathcal M_{s,p,n}$. The results in this paper are partially taken from the author's earlier arxiv post (Canonical blow-ups of grassmann manifolds, arxiv:2007.06200).

Theorems & Definitions (47)

• Definition 1.1
• Proposition 1.2: Proposition 1.5 in FW
• Theorem 1.3: Theorem 1.3 in FW
• Corollary 1.4
• Corollary 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Theorem 1.9
• Definition 2.3