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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)

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
06D05 Structure and representation theory of distributive lattices
06A07 Combinatorics of partially ordered sets
Full Text: DOI
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