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Gröbner bases of toric varieties. (English) Zbl 0714.14034
In this article projective toric varieties are studied from the viewpoint of Gröbner basis theory and combinatorics. We characterize the radicals of all initial ideals of a toric variety \(X_{{\mathcal A}}\) as the Stanley- Reisner ideals of regular triangulations of its set of weights \({\mathcal A}\). This implies that the secondary polytope \(\Sigma\) (\({\mathcal A})\) is a Minkowski summand of the state polytope of \(X_{{\mathcal A}}\). Here the lexicographic (resp. reverse lexicographic) initial ideals of \(X_{{\mathcal A}}\) arise from triangulations by placing (resp. pulling) vertices. We also prove that the state polytope of the Segre embedding of \({\mathbb{P}}^{r-1}\times {\mathbb{P}}^{s-1}\) equals the secondary polytope \(\Sigma (\Delta_{r-1}\times \Delta_{s-1})\) of a product of simplices.
Reviewer: B.Sturmfels

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
05B25 Combinatorial aspects of finite geometries
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