Torus quotient of the Grassmannian $G_{n,2n}$

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


Introduction
A polarized variety (X, L), where L is a very ample line bundle is said to be projectively normal if its homogeneous coordinate ring ⊕ m∈Z ≥0 H 0 (X, L ⊗m ) is integrally closed and it is generated as a C-algebra by H 0 (X, L) (see [2, Chapter II, Exercise 5.14]).For example, the projective line (P 1 , O(1)) is projectively normal.However, if we consider the rational twisted quartic curve in P 3 , i.e., image X = {[a 4 : a 3 b : ab 3 : b 4 ] ∈ P 3 : [a : b] ∈ P 1 } of the embedding i : P 1 ֒→ P 3 given by [a : b] → [a 4 : a 3 b : ab 3 : b 4 ], then (X, O X (1)) = (P 1 , O(3)) is normal but not projectively normal as the affine cone of X inside C 4 is not normal (see [2, Chapter I, Exercise 3.18]).
In [6], Kannan made an attempt to study projective normality of the GIT quotient of G 2,n by a maximal torus T of SL(n, C) with respect to the descent of O(n) (n is odd).There it was proved that the homogeneous coordinate ring of the GIT quotient of G 2,n by T with respect to the descent of O(n) is a finite module over the subring generated by the degree one elements.In [3], Howard et al. showed that the GIT quotient of G 2,n by T with respect to the descent of O( n 2 ) (respectively, O(n)) is projectively normal if n is even (respectively, if n is odd).In [14], Nayek et al. used graph theoretic techniques to give a short proof of the projective normality of the GIT quotient of G 2,n by T with respect to the descent of O(n) for any n.
To the best of our knowledge it is not known whether there is a suitable ample line bundle L on G r,n (r ≥ 3) such that the GIT quotient of G r,n by T with respect to the descent of the line bundle L is projectively normal (respectively, not projectively normal) with respect to the descent of L.
In this article, we prove the following: Theorem 1.1.The GIT quotient of G n,2n (n ≥ 3) by a maximal torus T of SL(2n, C) with respect to the descent of O(2) is not projectively normal (for more precise see Corollary 3.4).
The layout of the paper is as follows.In Section 2, we recall some preliminaries on algebraic groups, Standard Monomial Theory and Geometric Invariant Theory.In Section 3, we prove Theorem 1.1 (see Corollary 3.4).
Let V = C 2n and (e 1 , e 2 , . . ., e 2n ) be the standard basis of V.For a fixed integer r with 1 ≤ r ≤ 2n − 1, let G r,2n be the Grassmannian parameterizing the r-dimensional subspaces of C 2n .Then there is a natural projective variety structure on G r,2n given by the Plücker embedding π : G r,2n ֒→ P(∧ r V ) sending r-dimensional subspace to its r-th exterior power.The natural left action of SL(2n, C) on V induces an action of SL(2n, C) on ∧ r V and thus on P(∧ r V ), moreover, π is SL(2n, C)-equivariant.Let T be the maximal torus of SL(2n, C) consisting of diagonal matrices.Let O(1) denote the hyperplane line bundle on G r,2n given by the Plücker embedding π.Note that O(1) is SL(2n, C)-linearized, in particular, T -linearized.
Let I(r, 2n) denote the indexing set {i = (i 1 , i 2 , . . ., i r )|i j ∈ Z and 1 . ., i r ) ∈ I(r, 2n).Then {e i : i ∈ I(r, 2n)} forms a basis of ∧ r V. Let {p i : i ∈ I(r, 2n)} be the basis of the dual space (∧ r V ) * , which is dual to {e i : i ∈ I(r, 2n)}, i.e., p j (e i ) = δ ij .Note that p i 's are the i th Plücker coordinates of G r,2n .
In V, we fix a full flag For w = (w 1 , w 2 , . . ., w r ) in I(r, 2n), the Schubert variety in G r,2n associated to w is denoted by X(w) and is defined by , where 1 ≤ j ≤ 2n, 0 ≤ i ≤ r and w 0 := 0, w r+1 := 2n .
The definition of a Schubert variety X(w) depends on the choice of a full flag.However, given any two full flags i , which shows that X(w) is well defined up to an automorphism of V. We note that X(w) is a closed subvariety of G r,2n of dimension There is a natural partial order on I(r, 2n), given as follows: For w ∈ I(r, 2n), we also denote the restriction of the line bundle O(1) on G r,2n to X(w) by O(1).The monomial p τ 1 p τ 2 . . .p τm ∈ H 0 (X(w), O(m)), where τ 1 , τ 2 , . . ., τ m ∈ I(r, 2n) is said to be standard monomial of degree The standard monomials of degree m on X(w) form a basis of H 0 (X(w), O(m)).The Grassmannian G r,2n ⊆ P(∧ r V ) is precisely the zero set of the following well known Plücker relations: r+1 h=1 (−1) h p i 1 ,i 2 ,...,i r−1 j h p j 1 ,..., ĵh ,...,j r+1 , ( where {i 1 , . . ., i r−1 }, {j 1 , . . ., j r+1 } are two subsets of {1, 2, . . ., 2n} and ĵh means dropping the index j h . A point p ∈ X(w) is said to be semi-stable with respect to the T -linearized line bundle O(1) if there is a T -invariant section s ∈ H 0 (X(w), O(m)) for some positive integer m such that s(p) = 0. We denote the set of all semi-stable points of X(w) with respect to O(1) by ) and the stabilizer of p in T is finite.We denote the set of all stable points of X(w) with respect to O(1) by X(w) s T (O(1)).Let B (⊃ T ) be the Borel subgroup of SL(2n, C) consisting of upper triangular matrices.For 1 ≤ i ≤ 2n, define ε i : T → C × by ε i (diag(t 1 , . . ., t 2n )) = t i .Then S := {α i := ε i − ε i+1 | for all 1 ≤ i ≤ 2n − 1} forms the set of simple roots of SL(2n, C) with respect to T and B. Let {̟ i |i = 1, 2, . . ., 2n − 1} be the set of fundamental dominant weights corresponding to S.
For λ = m̟ r (m ≥ 1), we associate a Young diagram (denoted by Γ) with λ i number of boxes in the i-th column, where λ i := m for 1 ≤ i ≤ r.It is also called Young diagram of shape λ.A Young diagram Γ associated to λ is said to be a Young tableau if the diagram is filled with integers 1, 2, . . ., 2n.We also denote this Young tableau by Γ.A Young tableau is said to be standard if the entries along any column is non-decreasing from top to bottom and along any row is strictly increasing from left to right.Given a Young tableau Γ, let τ = {i 1 , i 2 , . . ., i r } be a typical row in Γ, where 1 ≤ i 1 < • • • < i r ≤ 2n.To the row τ , we associate the Plücker coordinate p i 1 ,i 2 ,...,ir .We set p Γ = τ p τ , where the product is taken over all the rows of Γ.Note that for w ∈ I(r, 2n), p Γ is a standard monomial on X(w) if Γ is standard and the bottom row of Γ is less than or equal to w.Further, p Γ is also called standard monomial on X(w) of shape λ.We use the notation p Γ and Γ interchangeably.Now we recall the definition of weight of a standard Young tableau Γ (see [12,Section 2,p.336]).For a positive integer 1 ≤ i ≤ 2n, we denote by c Γ (i), the number of boxes of Γ containing the integer i.The weight of Γ is defined as wt(Γ) We conclude this section by recalling the following key lemma about T -invariant monomials in

Main Theorem
First we recall that by [10, Theorem 3.10, p.764], 2 is the minimal integer k such that the line bundle O(k) on G n,2n descends to the GIT quotient T \\(G n,2n ) ss T (O(2)).In this section, we prove that there exists a Schubert subvariety X(v) of G n,2n admitting semi-stable points such that the GIT quotient T \\(X(v)) ss T (O( 2)) with respect to the descent of O( 2) is not projectively normal (see Theorem 3.3).As a consequence, we conclude that any Schubert variety X(w) containing X(v), the GIT quotient T \\(X(w)) ss T (O(2)) with respect to the descent of O( 2) is not projectively normal.In particular, T \\(G n,2n ) ss T (O(2)) is not projectively normal.
Now we consider the following w i 's such that w 1 ≤ w i for all 2 ≤ i ≤ 5 : Note that {w 1 , w 2 , w 3 , w 4 , w 5 } is precisely the set {w ∈ I(n, 2n) : w 1 ≤ w ≤ w 5 }.Further, note that w 2 and w 3 are non-comparable and w 2 , w 3 ≤ w 4 ≤ w 5 .Since w 1 ≤ w i and w Let us consider the following standard monomials Theorem 3.3.The GIT quotient X with respect to the descent of O( 2) is not projectively normal.
Proof.Consider the natural map f : R 1 ⊗ R 1 → R 2 of vector spaces given by X i ⊗ X j → X i X j for 1 ≤ i, j ≤ 5. Then f factors through second symmetric power S 2 R 1 of the vector space R 1 .For simplicity we also denote the factor map S 2 R 1 → R 2 by f.By Remark 3.2, we have dim(R 1 ) = 5 and dim(R 2 ) = 16.So, the map f : Corollary 3.4.The GIT quotient T \\(X(w)) ss T (O(2)) with respect to the descent of O(2) is not projectively normal for w ∈ I(n, 2n) such that w 5 ≤ w.In particular, the GIT quotient T \\(G n,2n ) ss T (O(2)) with respect to the descent of O(2) is not projectively normal.
Lemma 3.5.The homogeneous coordinate ring of X is generated by elements of degree at most two.
Proof.Let f ∈ R k be a standard monomial.We claim that f = f 1 f 2 , where f 1 is in R 1 or R 2 .The Young diagram associated to f has the shape (λ 1 , λ 2 , λ 3 , . . ., λ n ) = (2k, 2k, . . ., 2k n ).So the Young tableau Γ associated to f has 2k rows and n columns with strictly increasing rows and non-decreasing columns.Since f is T -invariant, by Lemma 2.1, we have c Γ (t) = k for all 1 ≤ t ≤ 2n.Let r i be the i-th row of the tableau.Let E i,j be the (i, j)-th entry of the tableau Γ and N t,j is the number of boxes in the j-th column of Γ containing the integer t.
By using (2.1), we have the following straightening laws in X(w 5 ) . Therefore, by using the above straightening laws we have .
By using ( * ) (see Appendix) we have Z Proof.Note that Y 2 = 0 on X(w i ) for all 1 ≤ i ≤ 4.
We claim that any relation among X i 's (1 ≤ i ≤ 4) given by a homogeneous polynomials of degree k is identically zero.Suppose where m = (m 1 , m 2 , m 3 , m 4 ) are tuples of non-negative integers such that m 1 +m 2 +m and c m 's are non-zero scalars.Then rewriting Eq. (3.1) as Recall that by Lemma 3.6, we have X Note that any monomial in X 1 , X 3 , X 4 , Y 1 (respectively, in X 1 , X 2 , X 4 , Y 1 ) is standard.Hence, c m = 0 for all m.Thus, X 1 , X 2 , X 3 , X 4 are algebraically independent.Further, by Lemma 3.5, and above surjectivity, the homogeneous coordinate ring of T \\(X(w 4 )) ss T (O(2)) is generated by X 1 , X 2 , X 3 , X 4 .On the other hand, by [1, Theorem 3.2.2,p.92], X(w 4 ) is normal.As T \\(X(w 4 )) ss T (O (2)) is an open subset of X(w 4 ), it is also normal.Hence, by Remark 3.1, the GIT quotient T \\(X(w 4 )) ss T (O(2)) with respect to the descent of O( 2) is projectively normal and isomorphic to Proj(C[X 1 , X 2 , X 3 , X 4 ]) = P 3 polarized with O(1).
, and a (m,n) 's are non-zero scalars.Now to prove that the homogeneous coordinate ring of T \\(X(w 5 )) ss T (O(4)) is generated by H 0 (X(w 5 ), O(4)) T as a C-algebra, it is enough to show that for each f as above and each monomial appearing in the expression of f is in the image of S k H 0 (X(w 5 ), O(4)) T , under the )).Then we have X = Proj(R), where R = k∈Z ≥0 R k and R k = H 0 (X(w 5 ), O(2k)) T .Note that R k 's are finite dimensional vector spaces.

Proposition 3 . 7 .
We have (i) The GIT quotient T \\(X(w 1 )) ss T (O(2)) with respect to the descent of O(2) is projectively normal and isomorphic to point.(ii)The GIT quotient T \\(X(w 2 )) ss T (O(2)) with respect to the descent of O(2) is projectively normal and isomorphic to P 1 polarized with O(1).(iii) The GIT quotient T \\(X(w 3 )) ss T (O(2)) with respect to the descent of O(2) is projectively normal and isomorphic to P 1 polarized with O(1).(iv) The GIT quotient T \\(X(w 4 )) ss T (O(2)) with respect to the descent of O(2) is projectively normal and isomorphic to P 3 polarized with O(1).