publication . Preprint . 2013

On singularities of lattice varieties

Mukherjee, Himadri;
Open Access English
  • Published: 28 Nov 2013
Toric varieties associated with distributive lattices arise as a fibre of a flat degeneration of a Schubert variety in a minuscule. The singular locus of these varieties has been studied by various authors. In this article we prove that the number of diamonds incident on a lattice point $\a$ in a product of chain lattices is more than or equal to the codimension of the lattice. Using this we also show that the lattice varieties associated with product of chain lattices is smooth.
arXiv: Mathematics::Algebraic GeometryHigh Energy Physics::Lattice
free text keywords: Mathematics - Combinatorics
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[1] G. Gratzer, Lattice theory: first concepts and distributive lattices, Dover pub. Inc., 2009.

[2] Daniel Kleitman,On Dedekind's problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc. 21 (1969), 677-682

[3] J. Brown and V. Lakshmibai, Singular loci of Bruhat-Hibi toric varieties, J. Alg., 319(2008) no 11. 4759-4799

[4] D. Eisenbud, Commutative algebra with a view toward Algebraic Geometry, Springer-Verlag, GTM, 150.

[5] D. Eisenbud and B. Sturmfels, Binomial ideals, preprint (1994).

[6] W. Fulton, Introduction to Toric Varieties, Annals of Math. Studies 131, Princeton U. P., Princeton N. J., 1993.

[7] N. Gonciulea and V. Lakshmibai, Degenerations of flag and Schubert varieties to toric varieties, Transformation Groups, vol 1, no:3 (1996), 215-248.

[8] N. Gonciulea and V. Lakshmibai, Singular loci of ladder determinantal varieties and Schubert varieties, J. Alg., 232 (2000), 360-395. [OpenAIRE]

[9] N. Gonciulea and V. Lakshmibai, Schubert varieties, toric varieties and ladder determinantal varieties , Ann. Inst. Fourier, t.47, 1997, 1013- 1064.

[10] R. Hartshorne, Algebraic Geometry, Springer-Verlag, New York, 1977.

[11] 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.

[12] J. Brown and V. Lakshmibai, Singular loci of Grassmann-Hibi toric varieties, Michigan Math. J. 59 (2010), no. 2, 243267

[13] G. Kempf et al, Toroidal Embeddings, Lecture notes in Mathematics,N0. 339, Springer-Verlag, 1973.

[14] V. Lakshmibai and H. Mukherjee, Singular loci of Hibi toric varieties,J Ramanujan Math. Soc., 26(2011) no. 1 (1-29)

[15] D. G. Wagner, Singularities of toric varieties associated with finite distributive lattices, Journal of Algebraic Combinatorics, no. 5 (1996) 149- 165 .

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