Torus quotient of the Grassmannian \(G_{n,2n}\) (Q6122531)

Let \(r\) and \(m\) be two positive integers such that \(1\leq r \leq m-1\). Let \(G_{r,m}\) denote the Grassmannian of all \(r\)-dimensional complex vector subspaces of \(\mathbb{C}^{m}\). Let \(T\) be the subgroup of all \(m\times m\)-diagonal matrices lying in \(SL(m, \mathbb{C})\). \(G_{r,m}\) is a homogeneous space for \(SL(m, \mathbb{C})\). Let \(\omega_{r}\) be the \(r\)-th fundamental weight. Let \(\mathcal{L}(\omega_{r})\) be the \(SL(m, \mathbb{C})\)-linearised ample line bundle on \(G_{r,m}\) corresponding to \(\omega_{r}\). The GIT quotient of \(G_{r,m}\) modulo \(T\) was studied by \textit{A. N. Skorobogatov} [Ann. Fac. Sci. Toulouse, Math. (6) 2, No. 3, 429--440 (1993; Zbl 0811.14020)] and \textit{S. Senthamarai Kannan} [Proc. Indian Acad. Sci., Math. Sci. 108, No. 1, 1--12 (1998; Zbl 0941.14018); Proc. Indian Acad. Sci., Math. Sci. 109, No. 1, 23--39 (1999; Zbl 1002.14017)]. Consider the special case \(r=n\) and \(m=2n\). So, we have the Grassmannian \(G_{n,2n}\) of all \(n\)-dimensional subspaces of \(\mathbb{C}^{2n}\). By a result of \textit{S. Kumar} [Transform. Groups 13, No. 3--4, 757--771 (2008; Zbl 1203.14051)], \(\mathcal{L}(2\omega_{n})\) descends to the GIT quotient of \(G_{n,2n}\) modulo \(T\). Let \(X\) be the GIT quotient of \(G_{n,2n}\) modulo \(T\). Let \(\mathcal{M}\) be the descent of \(\mathcal{L}(2\omega_{n})\) to \(X\). In the paper under review, the authors prove that the polarised variety \((X, \mathcal{M})\) is not projectively normal.