Torus quotient of the Grassmannian G n,2n

Abstract

Let G n,2n be the Grassmannian parameterizing the n-dimensional subspaces of ℂ 2n . The Picard group of G n,2n is generated by a unique ample line bundle 𝒪(1). Let T be a maximal torus of SL(2n,ℂ) which acts on G n,2n and 𝒪(1). By [10, Theorem 3.10, p. 764], 2 is the minimal integer k such that 𝒪(k) descends to the GIT quotient. In this article, we prove that the GIT quotient of G n,2n (n≥3) by T with respect to 𝒪(2)=𝒪(1) ⊗2 is not projectively normal when polarized with the descent of 𝒪(2).