The Cox ring of the space of complete rank two collineations
type: Item
instance of: scholarly article
title: The Cox ring of the space of complete rank two collineations
zbMATH Open document ID: 1255.14008
author: Jürgen Hausen
author: Michael Liebendörfer
published in: Mathematische Nachrichten
publication date: 2012-06-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.1171
Mathematics Subject Classification ID: 14C20
zbMATH DE Number: 6049555
zbMATH Keywords: Cox ring
zbMATH Keywords: torus action
zbMATH Keywords: Grassmannian
zbMATH Keywords: complete collineations
zbMATH Keywords: ambient modification
zbMATH Keywords: trinomials
zbMATH Keywords: Plücker relations
MaRDI profile type: Publication
OpenAlex ID: W1973669284
arXiv ID: 1110.1171
cites work: GIT-equivalence beyond the ample cone
cites work: The Cox ring of $\overline {M}_{0,6}$
cites work: Q4324883
cites work: Chow quotients of Grassmannian I
cites work: Q3835582
cites work: Complete collineations revisited
cites work: Complete collineations and blowing up determinantal ideals
DOI: 10.1002/MANA.201100043
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review text: The space \(X(u,c,d)\) of complete collineations is a compactification of the space of isomorphisms between \(u\)-dimensional subspaces of \(\mathbb C^c\) and \(\mathbb C^d\), respectively. For \(u=2\), Thaddeus has proven in 1995 that \(X(2,c,d)\) equals the blowing up of some GIT quotient of \(\text{Grass}(2,c+d)\) by a certain \(\mathbb C^*\)-action.NEWLINENEWLINEThe authors of the present paper use this description to determine the Cox ring of \(X(2,c,d)\) via so-called ambient modifications. This is a general method developed by the first author five years ago; its goal is to understand the behavior of Cox rings under blowing up of the base variety.NEWLINENEWLINEThe result is that \(\text{Cox}(X(u,c,d))\) is a \(\mathbb Z^3\)-graded ring which is explicitly given by a 1-parameter perturbation of the trinomial Plücker relations defining the affine cone over \(\text{Grass}(2,c+d)\).
reviewed by: Klaus Altmann
description: scientific article; zbMATH DE number 6049555