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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. GrĂ¤tzer’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
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