Nous construisons des variétés toriques normales (associées à des treillis distributifs finis) qui sont des grassmanniennes dégénérées. Nous déterminons aussi les lieux singuliers de ces variétés toriques, dans le cas où elles sont des variétés déterminantielles échelonnées (ladder determinantal varieties). Nous prouvons une version raffinée de la conjecture de Laksmibai & Sandhya [Criterion for smoothness of Schubert varieties in , Proc. Ind. Acad. Sci., 100 (1990), 45-52] sur les composantes du lieu singulier de certaines variétés de Schubert dans la variété des drapeaux.
We construct certain normal toric varieties (associated to finite distributive lattices) which are degenerations of the Grassmannians. We also determine the singular loci for certain normal toric varieties, namely the ones which are certain ladder determinantal varieties. As a consequence, we prove a refined version of the conjecture of Laksmibai & Sandhya [Criterion for smoothness of Schubert varieties in , Proc. Ind. Acad. Sci., 100 (1990), 45-52] on the components of the singular locus, for certain Schubert varieties in the flag variety.
@article{AIF_1997__47_4_1013_0,
author = {Gonciulea, Nicolae and Lakshmibai, Venkatramani},
title = {Schubert varieties, toric varieties and ladder determinantal varieties},
journal = {Annales de l'Institut Fourier},
pages = {1013--1064},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {47},
number = {4},
year = {1997},
doi = {10.5802/aif.1590},
mrnumber = {99a:14078},
zbl = {0878.14033},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.1590/}
}
TY - JOUR AU - Gonciulea, Nicolae AU - Lakshmibai, Venkatramani TI - Schubert varieties, toric varieties and ladder determinantal varieties JO - Annales de l'Institut Fourier PY - 1997 SP - 1013 EP - 1064 VL - 47 IS - 4 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1590/ DO - 10.5802/aif.1590 LA - en ID - AIF_1997__47_4_1013_0 ER -
%0 Journal Article %A Gonciulea, Nicolae %A Lakshmibai, Venkatramani %T Schubert varieties, toric varieties and ladder determinantal varieties %J Annales de l'Institut Fourier %D 1997 %P 1013-1064 %V 47 %N 4 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.1590/ %R 10.5802/aif.1590 %G en %F AIF_1997__47_4_1013_0
Gonciulea, Nicolae; Lakshmibai, Venkatramani. Schubert varieties, toric varieties and ladder determinantal varieties. Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1013-1064. doi : 10.5802/aif.1590. http://www.numdam.org/articles/10.5802/aif.1590/
[1] , Enumerative Combinatorics of Young tableaux, Marcel Decker, 1988. | Zbl
[2] , Combinatorial Theory, Springer-Verlag, New York, 1979. | MR | Zbl
[3] , , and , Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians, preprint (1997).
[4] , and , An introduction to Cohen-Macaulay partially ordered sets, I. Rival (editor), D. Reidel Publishing Company, 1982, 583-615. | MR | Zbl
[5] , Linear Algebraic Groups, second edition, Springer-Verlag, New York, 1991. | MR | Zbl
[6] , Groups et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968.
[7] , and , Ideals, Varieties and Algorithms, Springer-Verlag, New York, 1992.
[8] , Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. | Zbl
[9] and , Binomial ideals, preprint (1994). | Zbl
[10] , Introduction to Toric Varieties, Annals of Math. Studies, 131, Princeton U. P., Princeton N. J., 1993. | MR | Zbl
[11] and , Singularities of certain ladder determinantal varieties, Journal of Pure and Applied Algebra, 100 (1995), 59-75. | MR | Zbl
[12] and , Degenerations of flag and Schubert varieties to toric varieties, Transformation Groups, 1 (1996), 215-248. | MR | Zbl
[13] and , Singular loci of ladder determinantal varieties and Schubert varieties (preprint, submitted to Journal of Algebra). | Zbl
[14] , Algebraic Geometry, Springer-Verlag, New York, 1977. | MR | Zbl
[15] , Distributive lattices, affine semigroup rings, and algebras with straightening laws, Commutative Algebra and Combinatorics, Advanced Studies in Pure Math., 11 (1987) 93-109. | Zbl
[16] , Geometry of Coxeter groups, Pitman Adv. Pub. Prog. | Zbl
[17] et al, Toroidal Embeddings, Lecture notes in Mathematics, No. 339, Springer-Verlag, 1973. | MR | Zbl
[18] and , Multicones over Schubert varieties, Inv. Math., 87 (1987), 353-363. | EuDML | MR | Zbl
[19] , and , Cohomology of line bundles on G / B, Ann. E.N.S., t. 7 (1974), 89-137. | EuDML | Numdam | MR | Zbl
[20] and , Criterion for smoothness of Schubert varieties in SL(n) / B, Proc. Indian Acad. Sci. (Math. Sci.), 100 (1990), 45-52. | MR | Zbl
[21] and , Multiplicities of points in a minuscule G / P, Adv. Math., 84 (1990), 179-208. | MR | Zbl
[22] , Determinantal loci and the flag variety, Adv. Math., 74 (1989), 1-30. | MR | Zbl
[23] and , Projective normality of Flag varieties and Schubert varieties, Inv. Math., 79 (1985), 217-224. | EuDML | MR | Zbl
[24] , Schubert varieties are arithmetically Cohen-Macaulay, Inv. Math., 80 (1985), 283-294. | EuDML | MR | Zbl
[25] , Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. I.H.E.S., 65 (1987), 61-90. | EuDML | Numdam | MR | Zbl
[26] , Gröbner bases and convex polytopes, Lecture notes from the New Mexico State University, preprint, 1995. | MR | Zbl
[27] , Singularities of toric varieties associated with finite distributive lattices, preprint (1995). | MR | Zbl
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