×

zbMATH — the first resource for mathematics

Singularities of toric varieties associated with finite distributive lattices. (English) Zbl 0870.14037
For a finite lattice \(D\), consider the polynomial ring \(k[X_\alpha\mid \alpha\in D]\) over a field \(k\) in variables indexed by \(D\) with the usual grading \(\deg X_\alpha=1\) for all \(\alpha\in D\). With respect to the homogeneous binomial ideal \(I(D):= (X_\alpha X_\beta- X_{\alpha\wedge\beta} X_{\alpha\vee\beta}\mid \alpha,\beta\in D)\) consider the graded residue \(k\)-algebra \(k[D]:= k[X_\alpha\mid \alpha\in D]/I(D)\).
The associated projective variety \(V(D):= \text{Proj} (k[D])\) is known to be a (normal) toric variety if and only if the lattice \(D\) is distributive, by T. Hibi [in: Commutative algebra and combinatorics, US-Jap. joint Semin., Kyoto/Jap. 1985, Adv. Stud. Pure Math. 11, 93-109 (1987; Zbl 0654.13015)] in view of Birkhoff’s structure theorem and excluded sublattice theorem for finite distributive lattices, found, for instance, in G. Grätzer’s book [“General lattice theory” (1978; Zbl 0385.06015)] and in R. P. Stanley’s book [“Enumerative combinatorics”, Vol. I (1986; Zbl 0608.05001)]. According to Birkhoff’s structure theorem, a finite distributive lattice \(D\) is in categorical one-to-one correspondence with a finite bounded poset \(\widehat{P}\) (of join-irreducible elements in \(D\)).
The author describes the toric geometry of the toric projective variety \(V(D)\) directly in terms of the poset \(\widehat{P}\).
Reviewer: T.Oda (Sendai)

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
06D05 Structure and representation theory of distributive lattices
06A07 Combinatorics of partially ordered sets
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Benson, D. J.; Conway, J. H., Diagrams for modular lattices, J. Pure Appl. Algebra, 37, 111-116, (1985) · Zbl 0575.06008
[2] Dilworth, R. P.; Rival, I. (ed.), The role of order in lattice theory, (1982), Dordrecht, Boston
[3] D. Eisenbud and B. Sturmfels,“Binomial Ideals,” preprint. · Zbl 0873.13021
[4] Fulton, W., Introduction to toric varieties, (1993), Princeton, N. J.
[5] Geissinger, L., The face structure of a poset polytope, (1981), Barbados · Zbl 0523.06005
[6] G. Gratzer, General Lattice Theory, Birkhauser, Basel, Stuttgart, 1978.
[7] Hibi, T.; Nagata, M. (ed.); Matsumura, H. (ed.), Distributive lattices, affine semigroup rings, and algebras with straightening laws, No. 11, (1987), Amsterdam · Zbl 0654.13015
[8] Hibi, T., Hilbert functions of Cohen-Macaulay integral domains and chain conditions of finite partially ordered sets, J. Pure and Applied Algebra, 72, 265-273, (1991) · Zbl 0735.13009
[9] T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Publications, Glebe, Australia, 1992.
[10] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Annals of Math., 96, 318-337, (1972) · Zbl 0233.14010
[11] Lucchesi, C. L.; Younger, D. H., A minimax theorem for directed graphs, London Math. Soc. (2), 17, 369-374, (1978) · Zbl 0392.05029
[12] Stanley, R. P., Hilbert functions of graded algebras, Advances in Math., 28, 57-83, (1978) · Zbl 0384.13012
[13] Stanley, R. P., Two poset polytopes, Discrete Comput. Geom., 1, 9-23, (1986) · Zbl 0595.52008
[14] R. P. Stanley, Enumerative Combinatorics, vol. I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.
[15] Stanley, R. P., On the Hilbert function of a graded Cohen-Macaulay domain, J. Pure and Applied Algebra, 73, 307-314, (1991) · Zbl 0735.13010
[16] K. Vidyasankar, Some Covering Problems for Directed Graphs, Ph. D. Thesis, University of Waterloo, Ontario, 1976.
[17] D. G. Wagner, “Crowns, Cutsets, and Valuable Posets,” preprint. · Zbl 0866.06004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.