Journal ArticleDOI
Multi-cones over Schubert varieties
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In this paper, the authors give a criterion for a projective Cohen-Macaulay variety X to have rational singularity when X does not have a singularity, and they give an application to multi-cones based on Schubert varieties.Abstract:
Let ~q~ 1 . . . . , 2 ' , be invertible sheaves on a variety X. We may form the multigraded ring A = ~F(X, | where the vector (m) ranges in N". Then C = Spec(A) is a multi-cone. If X is a projective Cohen-Macaulay variety, n = l and L~'l is very ample, Serre has given a criterion for C to be Cohen-Macaulay in terms of the cohomology of sheaves on X. We know no reasonable analog of this result when n > 1. On the other hand we will give such a criterion for C to have rational singularity when X does. This is a stronger restriction on the singularities of C. Then we will give an application to multi-cones based on Schubert varieties. Let G be a reductive algebraic group, B a Borel subgroup and X ~G/B a Schubert variety. Let s176 &~ ..., 5q~ be the line bundles on G/B corresponding to the fundamental weights. An arbitrary line bundle corresponding to a dominant weight has the formread more
Citations
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Journal ArticleDOI
Degenerations of flag and Schubert varieties to toric varieties
N. Gonciulea,V. Lakshmibai +1 more
TL;DR: In this article, it was shown that determinantal varietes degenerate to (normal) toric varieties in a minusculeG/P, as well as the class of Kempf varieties in the flag varietySL(n)/B, to normal toric ones.
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The Cox Ring of a Del Pezzo Surface
Victor V. Batyrev,Oleg N. Popov +1 more
TL;DR: In this paper, the Cox ring of X r (r ⩾ 4) was shown to be similar to the homogeneous coordinate ring of the orbit of the highest weight vector in some irreducible representation of the algebraic group associated with the root system R r.
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Invariant prime ideals in quantizations of nilpotent Lie algebras
Milen Yakimov,Milen Yakimov +1 more
TL;DR: In this article, a family of subalgebras U + of a universal enveloping algebra Uq(g) associated to the elements of the corresponding Weyl group W is defined, and the corresponding poset is isomorphic to W, where H is the group of group-like elements of U q(g).
Journal ArticleDOI
Generic singularities of certain Schubert varieties
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TL;DR: In this paper, it was shown that the singularity of a maximal parabolic subgroup corresponding to a minuscule or cominuscule fundamental weight is isomorphic to the orbit closure of a highest weight vector in a certain Weyl module.
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Hilbert's 14th Problem and Cox Rings
TL;DR: In this article, it was shown that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants.
References
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Journal ArticleDOI
Frobenius splitting and cohomology vanishing for Schubert varieties
TL;DR: In this article, it was shown that the Cartier operator can give a manageable criterion for Frobenius splitting of Schubert varieties in characteristic p > 0, and that the map in cohomology H'(X, L) -+ H' (X, X, F*LP) = H'((X, LP), LP) is an injection.
Journal ArticleDOI
Projective normality of flag varieties and Schubert varieties
S. Ramanan,A. Ramanathan +1 more
Journal ArticleDOI
Equations defining schubert varieties and frobenius splitting of diagonals
A. Ramanathan,A. Ramanathan +1 more
TL;DR: The Grassmannian has a natural decomposition into affine spaces in projective 3-space as discussed by the authors, which is a projective variety with a natural embedding in P A' V, the projective space of lines in the r-th exterior power of a vector space V. For a vector in A" V to be decomposable its PRicker coordinates should satisfy certain quadratic relations.