Schubert varieties, toric varieties and ladder determinantal varieties
Gonciulea, Nicolae ; Lakshmibai, Venkatramani
Annales de l'Institut Fourier, Tome 47 (1997), p. 1013-1064 / Harvested from Numdam
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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 SL(n)/B, 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 SL(n)/B, Proc. Ind. Acad. Sci., 100 (1990), 45-52] on the components of the singular locus, for certain Schubert varieties in the flag variety.

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     author = {Gonciulea, Nicolae and Lakshmibai, Venkatramani},
     title = {Schubert varieties, toric varieties and ladder determinantal varieties},
     journal = {Annales de l'Institut Fourier},
     volume = {47},
     year = {1997},
     pages = {1013-1064},
     doi = {10.5802/aif.1590},
     mrnumber = {99a:14078},
     zbl = {0878.14033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 1013-1064. doi : 10.5802/aif.1590. http://gdmltest.u-ga.fr/item/AIF_1997__47_4_1013_0/

[1] S.S. Abhyankar, Enumerative Combinatorics of Young tableaux, Marcel Decker, 1988. | Zbl 0643.05001

[2] M. Aigner, Combinatorial Theory, Springer-Verlag, New York, 1979. | MR 80h:05002 | Zbl 0415.05001

[3] B.V. Batyrev, I. Ciocan-Fontanine, B. Kim and D. Van Straten, Conifold Transitions and Mirror Symmetry for Calabi-Yau Complete Intersections in Grassmannians, preprint (1997).

[4] A. Bjórner, A. Garsia and R. Stanley, An introduction to Cohen-Macaulay partially ordered sets, I. Rival (editor), D. Reidel Publishing Company, 1982, 583-615. | MR 83i:06001 | Zbl 0491.06005

[5] A. Borel, Linear Algebraic Groups, second edition, Springer-Verlag, New York, 1991. | MR 92d:20001 | Zbl 0726.20030

[6] N. Bourbaki, Groups et Algèbres de Lie, Chapitres 4, 5 et 6, Hermann, Paris, 1968.

[7] D. Cox, J. Little and D. O'Shea, Ideals, Varieties and Algorithms, Springer-Verlag, New York, 1992.

[8] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, 1995. | Zbl 0819.13001

[9] D. Eisenbud and B. Sturmfels, Binomial ideals, preprint (1994). | Zbl 0873.13021

[10] W. Fulton, Introduction to Toric Varieties, Annals of Math. Studies, 131, Princeton U. P., Princeton N. J., 1993. | MR 94g:14028 | Zbl 0813.14039

[11] D. Glassbrenner and K. E. Smith, Singularities of certain ladder determinantal varieties, Journal of Pure and Applied Algebra, 100 (1995), 59-75. | MR 96h:13029 | Zbl 0842.13008

[12] N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transformation Groups, 1 (1996), 215-248. | MR 98a:14065 | Zbl 0909.14028

[13] N. Gonciulea and V. Lakshmibai, Singular loci of ladder determinantal varieties and Schubert varieties (preprint, submitted to Journal of Algebra). | Zbl 1001.14019 | Zbl 01502160

[14] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977. | MR 57 #3116 | Zbl 0367.14001

[15] T. Hibi, Distributive lattices, affine semigroup rings, and algebras with straightening laws, Commutative Algebra and Combinatorics, Advanced Studies in Pure Math., 11 (1987) 93-109. | Zbl 0654.13015

[16] H. Hiller, Geometry of Coxeter groups, Pitman Adv. Pub. Prog. | Zbl 0483.57002

[17] G. Kempf et al, Toroidal Embeddings, Lecture notes in Mathematics, No. 339, Springer-Verlag, 1973. | MR 49 #299 | Zbl 0271.14017

[18] G. Kempf and A. Ramanathan, Multicones over Schubert varieties, Inv. Math., 87 (1987), 353-363. | MR 88c:14067 | Zbl 0615.14028

[19] V. Lakshmibai, C. Musili and C. S. Seshadri, Cohomology of line bundles on G / B, Ann. E.N.S., t. 7 (1974), 89-137. | Numdam | MR 50 #7175 | Zbl 0338.14017

[20] V. Lakshmibai and B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n) / B, Proc. Indian Acad. Sci. (Math. Sci.), 100 (1990), 45-52. | MR 91c:14061 | Zbl 0714.14033

[21] V. Lakshmibai and J. Weyman, Multiplicities of points in a minuscule G / P, Adv. Math., 84 (1990), 179-208. | MR 92a:14058 | Zbl 0729.14037

[22] S. B. Mulay, Determinantal loci and the flag variety, Adv. Math., 74 (1989), 1-30. | MR 90j:14071 | Zbl 0693.14021

[23] S. Ramanan and A. Ramanathan, Projective normality of Flag varieties and Schubert varieties, Inv. Math., 79 (1985), 217-224. | MR 778124 | MR 86j:14051 | Zbl 0553.14023

[24] A. Ramanathan, Schubert varieties are arithmetically Cohen-Macaulay, Inv. Math., 80 (1985), 283-294. | MR 788411 | MR 87d:14044 | Zbl 0541.14039 | Zbl 0555.14021

[25] A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Pub. Math. I.H.E.S., 65 (1987), 61-90. | Numdam | MR 908216 | MR 88k:14032 | Zbl 0634.14035

[26] B. Sturmfels, Gröbner bases and convex polytopes, Lecture notes from the New Mexico State University, preprint, 1995. | MR 1363949 | Zbl 0856.13020

[27] D. G. Wagner, Singularities of toric varieties associated with finite distributive lattices, preprint (1995). | MR 1382045 | Zbl 0870.14037