The Cox ring of the space of complete rank two collineations (Q2892948)

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 \(\text{Grass}(2,c+d)\).