Singular loci of Grassmann-Hibi toric varieties

Brown, Justin; Lakshmibai, V.

doi:10.1307/mmj/1281531454

Cited by 8 publications

(13 citation statements)

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“…Combining the above lemma with Theorem 4.6, we have the following The following fact from [4] (Lemmas 6.21, 6.22) holds for a general toric variety.…”

Section: Proofmentioning

confidence: 71%

“…Note that the result about the singular loci of B-H toric varieties of type A n (ω j ) was already proved in [4], as well as more results about the multiplicities of singular points, but using the unique combinatorics of these lattices.…”

Section: Minuscule Lattices a N−1 (ω J )mentioning

confidence: 98%

“…In [4], the authors conjectured that the above conjecture holds for other minuscule posets. The main result of this paper is the proof of the above conjecture for L being the Bruhat poset of Schubert varieties in a minuscule G/P (cf.…”

Section: Introductionmentioning

confidence: 94%

“…In [4], the authors gave a simple proof of the above conjecture for the Bruhat poset of Schubert varieties in the Grassmannian using just the combinatorics of the polyhedral cone associated to X(L).…”

Section: Introductionmentioning

confidence: 99%

“…It turns out that the above conjecture does not extend to a general Hibi toric variety X(L) (see Section 10 of [4] for a counterexample). In [4], the authors conjectured that the above conjecture holds for other minuscule posets.…”

Section: Introductionmentioning

confidence: 99%

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Singular loci of Bruhat–Hibi toric varieties

Brown

Lakshmibai

2008

Journal of Algebra

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For the toric variety X associated to the Bruhat poset of Schubert varieties in a minuscule G/P , we describe the singular locus in terms of the faces of the associated polyhedral cone. We further show that the singular locus is pure of codimension 3 in X, and the generic singularities are of cone type.Let K denote the base field which we assume to be algebraically closed of arbitrary characteristic. Given a distributive lattice L, let X(L) denote the affine variety in A #L whose vanishing ideal is generated by the binomials X τ X ϕ − X τ ∨ϕ X τ ∧ϕ in the polynomial algebra K[X α , α ∈ L] (here, τ ∨ ϕ (respectively τ ∧ ϕ) denotes the join-the smallest element of L greater than both τ, ϕ (respectively the meet-the largest element of L smaller than both τ, ϕ)). These varieties were extensively studied by Hibi in [9] where Hibi proves that X(L) is a normal variety. On the other hand, Eisenbud and Sturmfels show in [5] that a binomial prime ideal is toric (here, "toric ideal" is in the sense of [14]). Thus one obtains that X(L) is a normal toric variety. We shall refer to such a X(L) as a Hibi toric variety.For L being the Bruhat poset of Schubert varieties in a minuscule G/P , it is shown in [7] that X(L) flatly deforms to G/P (the cone over G/P ), i.e., there exists a flat family over A 1 with

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“…Combining the above lemma with Theorem 4.6, we have the following The following fact from [4] (Lemmas 6.21, 6.22) holds for a general toric variety.…”

Section: Proofmentioning

confidence: 71%

Section: Minuscule Lattices a N−1 (ω J )mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations