Projective normality of torus quotients of flag varieties

Arpita, Nayek,; Pattanayak, Santosha Kumar; Jindal, Shivang

doi:10.1016/j.jpaa.2020.106389

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Torus quotient of the Grassmannian G n,2n

Nayek,

Saha

2023

Comptes Rendus. Mathématique

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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). Résumé. SoitG n,2n la Grassmannienne des sous-espaces de dimension n de C 2n . Le groupe de Picard de G n,2n est engendré par un unique fibré en droites ample O (1). Fixons un tore maximal T du groupe SL(2n, C) qui agit sur G n,2n et O (1). D'après [10, Theorem 3.10, p. 764], 2 est l'entier minimal k tel que O (k) descende au quotient GIT. Dans cet article, nous prouvons que le quotient GIT de G n,2n (n ≥ 3) par T par rapport à O (2) = O (1) ⊗2 n'est pas projectivement normal lorsqu'il est polarisé avec la descente de O (2).

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Section: Introductionmentioning

confidence: 99%

Torus quotient of the Grassmannian G n,2n

Nayek,

Saha

2023

Comptes Rendus. Mathématique

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“…In [1], the authors used combinatorial method to show that T \\(G 2,n ) ss T (L(nω 2 )) is projectively normal with respect to the descent of the line bundle L(nω 2 ) for n is odd. In [19], the authors used graph theoretic technique to give a short proof of the projective normality of T \\(G 2,n ) ss T (L(nω 2 )) for general n. In [10], authors studied the torus quotients of Schubert varieties in orthogonal grassmannians. Authors showed the following: when G = Spin(8, C) the GIT quotient of G/P α4 is projectively normal with respect to the descent of the ample line bundle L(2ω 4 ) and isomorphic to the projective space P 2 .…”

Section: Introductionmentioning

confidence: 99%

Torus quotients of Schubert varieties in the Grassmannian $G_{n,2n}$

Arpita¹,

Saha²

2021

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Let G = SL(2n, C) (n ≥ 3) and T be a maximal torus in G. In this article, we study the GIT quotients of the Schubert varieties in G n,2n for the action of T. We show that there exists a Schubert variety X(v) in G n,2n admitting semistable point such that the GIT quotient of X(v) is not projectively normal with respect to the descent of the ample line bundle L(2ωn). As a consequence, we conclude that for any Schubert variety X(w) in G n,2n containing X(v), the GIT quotient of X(w) is not projectively normal with respect to the descent of the ample line bundle L(2ωn). In particular, the GIT quotient of G n,2n (n ≥ 3) is not projectively normal with respect to the descent of the ample line bundle L(2ωn).

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